On graded division rings

We develop the theory of group graded division ring parallel to the one by P. Cohn for (ungraded) division rings.


Introduction
Let R be a commutative ring. It is well known the prime ideals or R classify the homomorphisms from R to division rings. Indeed, for any prime ideal P of R, we obtain a homomorphism from R to a division ring via the natural homomorphism R → Q(R/P ), where Q(R/P ) denotes the field of fractions of R/P . Conversely, if ϕ : R → D is a homomorphism from R to a division ring D, then P = ker ϕ is a prime ideal of R, ϕ factors through R → Q(R/P ) and therefore the division subring of D generated by the image of R is R-isomorphic to Q(R/P ). Moreover, let P ⊆ P ′ be prime ideals of R. The localization of R/P at the prime ideal P ′ /P yields a local subring of Q(R/P ) with residue field isomorphic to Q(R/P ′ ). This implies that any fraction ab −1 ∈ Q(R) which is defined in Q(R/P ′ ) it is also defined in Q(R/P ). Also, looking at the determinants of matrices, one sees that any matrix with entries in R that becomes invertible in Q(R/P ′ ) then it also becomes invertible in Q(R/P ).
If the ring R is not commutative, prime ideals no longer classify the homomorphisms to division rings. It may even possible that R has infinitely many different "fields of fractions", see for example [18,Section 9]. Let R be any ring. An epic R-division ring is a ring homomorphism R → K where K is a division ring generated by the image of R. In [6], P. M. Cohn showed that the epic R-division rings are characterized up to R-isomorphism by the collection of square matrices over R which are carried to matrices singular over K. This set of matrices is called the singular kernel of R → K. He also gave the precise conditions for a set of square matrices over R to be a singular kernel, calling such a collection a prime matrix ideal of R. The name comes from the fact that, if we endow the set of square matrices over R with certain two operations of sum and product (one of them partial), those sets have a similar behaviour to prime ideals. These operations are defined so that, when defined on square matrices over a commutative ring, the determinant of the sum of matrices equals the sum of the determinants and the determinant of a product of matrices equals the product of the determinants. Also in [6], Cohn showed that if P, P ′ are prime matrix ideals of R and R → K P , R → K P ′ are the corresponding epic R-division rings, then P ⊆ P ′ if and only if there exists a local subring of K P containing the image of R with residue class division ring R-isomorphic to K P ′ . We say that there exists a specialization from K P to K P ′ . Furthermore, if a rational expression built up from elements of R makes sense in K P ′ , then it can also be evaluated in K P . P. M. Cohn also provided conditions on square matrices over R equivalent to the existence of (injective) homomorphisms from R to division rings and to the existence of a best epic R-division ring in the sense that a rational expression that makes sense in some epic R-division ring, makes sense in it.
In [24], P. Malcolmson described several alternative ways of determining epic R-division rings. One of them is induced from the notion of rank of a matrix over a division ring. If ϕ : R → K is an epic R-division ring, we can associate to each matrix over R the rank of this matrix when considered over K via ϕ. He determined which functions from the set of matrices over R with values in N are rank functions induced from epic R-division rings. Another alternative way of determining epic Rdivision rings described by P. Malcolmson is induced from the notion of dimension over a division ring. More precisely, if R → K is an epic R-division ring, we can associate with each finitely presented right R-module M the number dim K (M ⊗ R K) ∈ N. He described which functions from the class of finitely presented right R-modules with values in N are induced from epic R-division rings as dimensions. Another important feature of rank functions is that, theoretically speaking, it is easy to know when there exists a specialization from an epic R-division ring to another in terms of rank functions as defined by P. Malcolmson. In [28], A. Schofield gave another equivalent notion to that of epic R-division rings in terms of a rank function that satisfies certain natural conditions. This time it is a function from the class of homomorphisms between finitely generated projective right R-modules with values in N. We would like to remark that Sylvester rank functions with values in R + have proved useful in many different situations [1,2,9,10,11,15,16,28].
The theory of group graded rings has played an important role in Ring Theory (see for example [12], [26]) and many of the results in classical ring theory have a mirrored version for group graded rings. Furthermore, if R is a filtered ring, it has proved fruitful to study the associated graded ring, which usually is a simpler object, in order to obtain information about the original ring.
The main aim of this article is to develop Cohn's theory on division rings in the context of group graded rings. More precisely, let Γ be a group and R = γ∈Γ R γ be a Γ-graded ring. A Γ-graded epic R-division ring is a homomorphism of Γgraded rings R → K where K is a Γ-graded division ring generated by the image of R. Matrices over R represent homomorphisms between finitely generated free Rmodules. Homomorphisms of Γ-graded modules between Γ-graded free R-modules are given by (what we call) homogeneous matrices. These are m × n matrices A for which there exist α 1 , . . . , α m , β 1 , . . . , β n ∈ Γ such that each (i, j) entry of A belongs to R αiβ −1 j . We show that Γ-graded epic R-division rings R → K are characterized, up to R-isomorphism of Γ-graded rings, by the collection of homogeneous matrices which are carried to singular matrices over K. These sets are called the gr-singular kernel of R → K. We give the precise conditions under which a collection of homogeneous matrices over R is a gr-singular kernel and thus defining the concept of gr-prime matrix ideal. If P, P ′ are gr-prime matrix ideals of R and R → K P , R → K P ′ are the corresponding Γ-graded epic R-division rings, then P ⊆ P ′ if and only if there exists a Γ-graded local subring of K P that contains the image of R with residue class Γ-graded division ring R-isomorphic to K P ′ as Γ-graded rings. Furthermore, if a homogeneous rational expression obtained from elements of R make sense in K P ′ then it can also be evaluated in K P . We then provide conditions on the set of square homogeneous matrices over R that characterize when there exists an (injective) homomorphism of Γ-graded rings from R to a Γ-graded division ring and when there exists a best Γ-graded epic R-division ring. We also provide the graded concepts corresponding to the different rank functions defined by Malcolmson and Schofield. We show they give alternative ways of determining Γgraded epic R-division rings in terms of rank functions from the set of homogeneous matrices, from the class of Γ-graded finitely presented modules and from the class of Γ-graded homomorphisms between Γ-graded projective R-modules, respectively, all of them with values in N.
In the study of division rings, one of the pioneering works carrying the information from the associated graded ring to the original filtered ring was [5]. P. M. Cohn showed that if a ring R endowed with a valuation with values in Z is such that its associated graded ring is a (graded) Ore domain, then R can be embedded in a division ring. Other proofs of this result can be found in [20] and in [3] together with [19]. More recently, a generalization of the result by Cohn has been given by A. I. Valitskas [29]. We believe that our work could be helpful in order to generalize the result by Cohn to a greater extent than has been done by Valitskas. An elementary application of our theory is as follows. Suppose that R is a ring graded by a group Γ. As an immediate consequence of [26,Proposition 1.2.2], one obtains that if there exists an (injective) homomorphism from R to a division ring, then there exists an (injective) homomorphism of Γ-graded rings from R to a Γ-graded division ring. Thus if one shows that there do not exist (injective) homomorphisms of Γ-graded rings from R to Γ-graded division rings, then there do not exist (injective) homomorphisms from R to division rings. See section 9 for other similar results.
We end this introduction by showing that the existence of an (injective) homomorphism from a Γ-graded ring R to a division ring is not equivalent to the existence of a homomorphism of Γ-graded rings from R to a Γ-graded division ring. For that we produce an easy example of a graded ring for which there does not exist a homomorphism to a division ring but it is embeddable in a graded division ring. Let T be the ring obtained as localization of Z at the prime ideal 3Z. Let R be the ring T [i] ⊆ C. Let C 2 = x be the cyclic group of order two, and let σ : C 2 → Aut(R) be the homomorphism of groups which sends x to the automorphism induced by the complex conjugation. Set now S = R[C 2 ; σ]. That is, S is the skew group ring of G over R induced by σ. Hence S is a C 2 -graded ring, S = S e + S x where S e = R and S x = Rx and the product is determined by xr = rx for all r ∈ R. Clearly S is embeddable in the C 2 graded division ring Q[C 2 ; σ]. Suppose that there exists a homomorphism of rings from S to a division ring K. Let ϕ : S → K be such homomorphism. Since (1 − x)(1 + x) = 0, then either ϕ(1 + x) = 0 or ϕ(1 − x) = 0. If ϕ(1 + x) = 0, then 0 = ϕ(1 + x) = 1 + ϕ(x). Thus ϕ(x) = −1. But then (−1)ϕ(i) = ϕ(xi) = ϕ(−ix) = −ϕ(i)(−1) = ϕ(i). Since ϕ(i) = 0, then K has characteristic two. This is a contradiction because ϕ induces a homomorphism from R = S e to K and 2 is invertible in R. In the same way, it can be shown that if ϕ(1 − x) = 0, then ϕ(x) = 1 and, again, it implies that the characteristic of K is 2, a contradiction.
In Section 1, we introduce some of the notation that will be used throughout the paper and provide a short survey about the results on graded rings that will be used.
Let Γ be a group. A Γ-almost graded division ring is a (not necessarily graded) homomorphic image of a Γ-graded division ring. For example, let K be a field and consider the group ring K[Γ]. It is a Γ-graded division ring, and the augmentation map K[Γ] → K, which is not a homomorphism of Γ-graded rings, endows K with as structure of Γ-almost graded division ring. In the nongraded context, this concept is not necessary because a nontrivial image of a division ring is again a division ring. In Section 2, we show that if R is a Γ-graded ring, ϕ : R → D is a homomorphism of Γ-graded rings with D a Γ-graded division ring and ψ : D → E is a ring homomorphisms where E is a nonzero ring, then the homogeneous matrices over R that become invertible via ϕ and via ψϕ are the same. Thus (a posteriori) ψ(D) determines a Γ-graded epic R-division ring.
The main results in Section 3 are as follows. Let ϕ : R → D be a homomorphism of Γ-graded rings and let Σ be a set of square homogeneous matrices with entries in R. Suppose that the matrices of Σ become invertible in D via ϕ. Then, under certain natural conditions on Σ, the entries of the inverses of the matrices in Σ are the homogeneous elements of a Γ-graded subring of R. Moreover, if D is a Γ-graded division ring and Σ the set of homogeneous matrices that become invertible under ϕ, then any homogeneous element of D is an entry of the inverse of some matrix in Σ.
Section 4 begins showing that the universal localization R Σ of the Γ-graded ring R at a set of homogeneous matrices is again a Γ-graded ring. Then it is shown that a homomorphism of Γ-graded rings ϕ : R → D, where D is Γ-graded division ring, is an epimorphism in the category of Γ-graded rings if and only if D is generated by the image of ϕ. If this is the case, we say that (K, ϕ) is a Γ-graded epic R-division ring and we prove that if Σ is the set of square homogeneous matrices that become invertible in D via ϕ, then R Σ is a Γ-graded local ring with Γ-graded residue division ring R-isomorphic to D. Then the concept of gr-specialization between Γ-graded epic R-division rings is defined. The section ends showing that the existence of a gr-specialization from (K, ϕ) to another Γ-graded epic R-division ring (K ′ , ϕ ′ ) is equivalent to say that all the homogeneous rational expressions (from elements of R) that make sense in (K ′ , ϕ ′ ) make sense in (K, ϕ) too, and that it is also equivalent to fact that any homogeneous matrix over R that becomes invertible in (K ′ , ϕ ′ ) becomes invertible in (K, ϕ) too.
Section 5 is devoted to the proof of the graded version of the so called Malcolmson's criterion [25] and an important consequence. This criterion determines the kernel of the natural homomorphism from R to the universal localization R Σ of R at certain sets of homogeneous matrices. As a corollary one obtains a sufficient condition for the ring R Σ not to be the zero ring. Both results play a key role in the following section, but the proof of Malcolmson's criterion is very long and technical.
The concept of gr-prime matrix ideal is given in Section 6 and it is shown that the different Γ-graded epic R-division rings are determined by the gr-prime matrix ideals up to R-isomorphism of Γ-graded rings.
In Section 7, the concepts of a gr-matrix ideal and of the radical of a gr-matrix ideal are defined and it is characterized how is the gr-matrix ideal generated by a set of homogeneous square matrices. Then it is proved that gr-prime matrix ideals behave like prime ideals in a commutative ring. All these concepts are used to provide necessary and sufficient condition for the existence of homomorphisms (embeddings) of Γ-graded rings to Γ-graded division rings.
The basic theory of Sylvester rank functions in the graded context with values in N is developed in Section 8. The main difference with the ungraded case stems from the the fact that, in the graded case, the same homogeneous matrix can define more than one homomorphism between Γ-graded free modules. As far as we know, this is the first paper where Sylvester rank functions are considered for graded objects.
In Section 9, we deal with a new situation that appears in the graded context. If Γ is a group and R is a Γ-graded ring, then the ring R can be considered as a Γ/Ω-graded ring for any normal subgroup Ω of Γ. Thus there are Γ-graded and Γ/Ω-graded versions of the concepts studied before. In this section, we try to relate them. Note that when Ω = Γ, a Γ/Ω-graded epic R-division ring is simply an R-division ring and thus one can relate the theory of Γ-graded division rings and the theory of division rings as developed by Cohn. The last section is devoted to identify inverse limits in the category of Γ-graded epic R-division rings with specializations as morphisms with certain ultraproducts of Γ-graded epic R-division rings. In the context of division rings, a similar result was given in [21, Section 7], but our proof is more direct and general even when specialized to the ungraded case.
A second paper is in the works where we deal, among other topics, with the graded versions of weak algorithm, (semi)firs and (pseudo-)Sylvester domains.
We would like to finish this introduction by pointing out that most of the techniques used in this paper are adaptations of the ones from the works by P. M. Cohn and P. Malcolmson. We just take credit for realizing that they can be applied in the more general setting of group graded rings.

Basic definitions and notation
Rings are supposed to be associative with 1. We recall that a domain is a nonzero ring such that for elements x, y of the ring, the equality xy = 0 implies that either x = 0 or y = 0. A division ring is a nonzero ring such that every nonzero element is invertible. For a ring R, we define M(R) the set of all square matrices of any size. Also, for each i with 1 ≤ i ≤ n, let e i denote the column  in which the i-th entry is 1 and the other entries are zero. Let A ∈ M n (R). We say that A is full if whenever A = P Q, with P ∈ M n×r (R) and Q ∈ M r×n (R), then r ≥ n. If we think of A as an endomorphism of the free (right) R-module R n , it means that A does not factor through R r with r < n. We say that A is hollow if it has an r × s block of zeros where r + s > n. It is well known that a hollow matrix is not full.
Let S be a ring and f : R → S be a ring homomorphism. For each matrix M with entries in R, we denote by M f the matrix whose entries are the images of the entries of M by f , that is, if a ij ∈ R is the (i, j)-entry of M , then the (i, j)-entry of M f is f (a ij ). Given a set of matrices Σ, we denote Σ f = {M f : M ∈ Σ}. We say that the ring homomorphism f : We proceed to give some basics on group graded rings that can be found in [26] and [12], for example.
If Γ is a group, the identity element of Γ will be denoted by e.
is called the set of homogeneous elements of R. It is well known that the identity element 1 ∈ R belongs to R e , that R e is a subring of R and that if , then the ideal of R generated by X is a graded ideal. If I is a graded ideal, then the quotient ring R/I is a Γ-graded ring with R/I = γ∈Γ (R/I) γ , where (R/I) γ = (R γ + I)/I. A Γ-graded domain is a nonzero Γ-graded ring such that if x, y ∈ h(R), the equality xy = 0 implies that either x = 0 or y = 0. A Γ-graded division ring is a nonzero Γ-graded ring such that every nonzero homogeneous element is invertible. A commutative Γ-graded division ring is a Γ-graded field. Clearly, any Γ-graded division ring is a Γ-graded domain.
A Γ-graded ring R is called a Γ-graded local ring if the two-sided ideal m generated by the noninvertible homogeneous elements is a proper ideal. In this case, the Γ-graded ring R/m is a Γ-graded division ring and it will be called the residue class Γ-graded division ring of R.
For Γ-graded rings R and S, a homomorphism of Γ-graded rings f : R → S is a ring homomorphism such that f (R γ ) ⊆ S γ for all γ ∈ Γ. An isomorphism of Γ-graded rings is a homomorphism of Γ-graded rings which is bijective. Notice that the inverse is also an isomorphism of Γ-graded rings.
Let Ω be a normal subgroup of Γ. Consider the Γ-graded ring R = γ∈Γ R γ . It can be regarded as a Γ/Ω-graded ring as follows For Γ-graded R-modules M and N , a homomorphism of Γ-graded R-modules f : M → N is a homomorphism of R-modules such that f (M γ ) ⊆ N γ for all γ ∈ Γ. In this case, ker f is a graded submodule of M and Im f is a graded submodule of N .
with a homogeneous basis. It is well known that the Γ-graded free R-modules are of the form i∈I R(δ i ), where I is an indexing set and δ i ∈ Γ.
A Γ-graded R-module P is called a Γ-graded projective module if for any diagram of Γ-graded R-modules and homomorphisms of Γ-graded modules there is a graded R-module homomorphism h : P → M with gh = u. As in the ungraded case, the following statements are equivalent ways of saying that P is a Γ-graded projective module (1) P is Γ-graded and projective as an R-module.
(2) Every short exact sequence of homomorphisms of Γ-graded R-modules 0 → L → M → P → 0. splits via a homomorphism of Γ-graded R-modules. (3) P is isomorphic, as Γ-graded R-module, to a direct summand of a Γ-graded free R-module.
Let P be a Γ-graded projective R-module and let Ω be a normal subgroup of Γ. If we regard P as a Γ/Ω-graded R-module, then P is also projective as a Γ/Ω-graded R-module.
When m = n, we will write M n (R)[α][β] and M n (R). The set of all such matrices will be denoted by M (R), that is, , then r ≥ n. If we think of A as a homomorphism of Γ-graded modules between two Γ-graded free R-modules, it means that for all α, β ∈ Γ n , such that A defines a graded homomorphism R n (β) → R n (α), then it never factors by any graded homommorphism R n (β) → R r (λ) with r < n.
Suppose that A ∈ M n (R)[α][β], E ∈ M n (R) is a permutation matrix obtained permuting the rows of I n according to the permutation σ ∈ S n . Then Hence, for permutation matrices E, F of appropriate size, a matrix A ∈ M(R) is gr-full if, and only if, EAF is gr-full.
A hollow matrix A ∈ M(R) is not gr-full. Indeed, suppose that A has an r × s block of zeros. There exist permutation matrices E, F such that EAF = ( T 0 U V ), that is, the block of r × s zeros is in the north-east corner. Then for some sequences α, β, γ of elements of Γ. The result now follows because Let D be a Γ-graded division ring and M be a Γ-graded D-module. As in the ungraded case, the following assertions hold true (1) Any Γ-graded D-module is graded free.
(2) Any D-linearly independent subset of M consisting of homogeneous elements can be extended to a homogeneous basis of M . We remark that, over a Γ-graded division, the concepts of gr-full matrix and of invertible matrix coincide.
Let D be a Γ-graded division ring. Let A ∈ M m×n (D). The elementary homogeneous row (column) operations on A are (1) Interchange two rows (columns) of A.
(2) Multiply a row on the left (a column on the right) by a nonzero homogeneous element. Notice that those three operations on the rows (columns) can be obtained multiplying A on the left (right) by an invertible matrix in The rank of A is the dimension of the right D-module spanned by its columns. The matrix A can be regarded as a D-linear map of right D-modules R(β 1 ) ⊕ · · · ⊕ R(β n ) → R(α 1 ) ⊕ · · · ⊕ R(α n ). The rank of A coincides with the dimension of the image of A. The rank of A can also be computed reducing the matrix A to column echelon form by homogeneous column operations. It is the number of nonzero columns of the column echelon form.
The rank of A equals also the dimension of the left free D-module spanned by its rows. The matrix A can be regarded as a D-linear map of left D-modules The rank of A coincides with the dimension of the image of A. The rank of A can also be computed reducing the matrix A to row echelon form by homogeneous row operations. It is the number of nonzero rows of the row echelon form.
Furthermore, the rank of A coincides with the size of a largest invertible square submatrix (obtained by eliminating rows and/or columns). We will denote the rank of A by rank(A).

Almost graded division rings
Throughout this section, let Γ be a group. We say that a ring R is a Γ-almost graded ring if there is a family {R γ : γ ∈ Γ} of additive subgroups R γ of R such that 1 ∈ R e , R = γ∈Γ R γ and R γ R γ ′ ⊆ R γγ ′ for all γ, γ ′ ∈ Γ. The name of almost graded rings was chosen to be compatible with the definition of almost strongly graded rings given in [26, p.14]. We define supp R = {γ ∈ Γ : R γ = {0}}. Given two Γ-almost graded rings R and S, a ring homomorphism f : R → S is a homomorphism of Γ-almost graded rings if f (R γ ) ⊆ S γ for all γ ∈ Γ. Clearly, any Γ-graded ring R = γ∈Γ R γ is a Γ-almost graded ring in the natural way. Given two Γ-graded rings R, S, a homomorphism of Γ-almost graded rings is in fact a homomorphism of Γ-graded rings. Let R be a Γ-graded ring, S be ring and f : R → S be a ring homomorphism. Then Im f is a Γ-almost graded ring with (Im f ) γ = f (R γ ) and the restriction f : R → Im f is a homomorphism of Γ-almost graded rings. Furthermore, any Γ-almost graded ring can be regarded in this way. More precisely, suppose that R = γ∈Γ R γ is a Γ-almost graded ring. Set R γ to be a disjoint copy of R γ . If a ∈ R γ , denote byã ∈ R γ the disjoint copy of a ∈ R γ . Consider the Γ-graded additive group R = γ∈Γ R γ . Define R γ × R γ ′ → R γγ ′ by (ã,b) → ab, and extend it by distributivity to R × R → R. This endows R with a structure of Γ-graded ring such that supp R = supp R and ϕ : a → a for all a ∈ R γ , γ ∈ Γ, is a homomorphism of Γ-almost graded rings such that ϕ ( R γ Another important example is as follows. If S = α∈Γ/Ω S α is a Γ/Ω-graded ring, then S can be endowed with a structure of Γ-almost graded ring defining S γ = S α for all γ ∈ α. Suppose that R = γ∈Γ R γ is a Γ-graded ring and that that Ω is a normal subgroup of Γ. The ring R is a Γ/Ω-graded ring defining R α = γ∈α R γ for each α ∈ Γ/Ω. If f : R → S is a homomorphism of Γ/Ω-graded rings, then it is a homomorphism of Γ-almost graded rings. Let Γ ′ be the subgroup of Γ generated by supp R. Observe that if Ω is a normal subgroup of Γ ′ (instead of Γ), S is a Γ ′ /Ω-graded ring and f : R → S a homomorphism of Γ ′ /Ω-graded rings, then f is a homomorphism of Γ ′ -almost graded rings.
We say that a nonzero ring E is a Γ-almost graded division ring if E is a Γalmost graded ring such that every nonzero element The following easy result tells us that Γ-almost graded division rings are graded division rings although not necessarily of type Γ.
Lemma 2.1. Let E be a Γ-almost graded division ring. The following assertions hold true.
The other part is analogous. Thus (1) is proved.
(2) is a consequence of (1). Since 1 ∈ E e , then (3) follows from (2). (4) First note that, for each γ ∈ Γ, the condition E γ ∩ E e = {0} implies that And the result is proved. Let R be a Γ-graded ring, S be a ring and f : R → S be a ring homomorphism.
If DC(f ) = S and DC(f ) is a Γ-almost graded divison ring, we say that S is the Γ-almost graded division ring generated by Im f .
Notice also that if S is a division ring, then DC(f ) is a Γ-almost graded division ring.
Note that if S is a Γ-graded ring, and f : R → S is a homomorphism of Γ-graded rings, then (S n ) γ ⊆ S γ for each n ≥ 0. Therefore (DC(f )) γ ⊆ S γ and DC(f ) is a Γ-graded subring of S. It is the least subring of S that contains Im f and is closed under inversion of homogeneous elements. Moreover if S is a Γ-graded division ring, then DC(f ) is a Γ-graded division subring of S. In this case, if S = DC(f ) we say that S is the Γ-graded division ring generated by Im f . Proof.

Graded rational closure
Throughout this section, let Γ be a group.
We begin this section introducing some important notation that will be used throughout.
For each A ∈ M n (R), the last column will be called A ∞ and the matrix consisting of the remaining n − 1 columns will be called A • . We will write A = (A • A ∞ ).
If A ∈ M n×(n+1) (R), we will denote by A 0 its first column, by A ∞ its last column and by A • the matrix consisting of the other n − 1 columns, that is, we will write A = (A 0 A • A ∞ ). We will call the matrix (A 0 A • ) the numerator of A and the matrix ( Again, we remark that if n = 1, then A • , β • , u • are empty and thus A = (A 0 A ∞ ), β = (β 0 , β ∞ ) and u = ( u0 u∞ ).
We say that the subset Σ of M(R) is gr-lower semimultiplicative if it satisfies the following two conditions: (i) (1) ∈ Σ, i.e. the identity matrix of size 1 × 1 belongs to Σ.
An gr-upper semimultiplicative subset of M(R) is defined analogously. A subset Σ of M(R) is gr-multiplicative if it satisfies the following two conditions (i) Σ is lower gr-semimultiplicative. (ii) If A ∈ Σ, then EAF ∈ Σ for any permutation matrices E, F of appropriate size.
Remark 3.1. We remark that if Σ is gr-multiplicative then it is also an upper gr-semimultiplicative subset of M (R).
Then, since Σ is lower gr-semimultiplicative, Notice that if E, F are permutation matrices, then E f , F f are also permutation matrices.
Note that if S is a Γ-graded ring, f : R → S is a graded homomorphism and , and the (j, i)-entry of (A f ) −1 belongs to R βj α −1 i . With this in mind, we make the following definition.
Let R = γ∈Γ R γ be a Γ-graded ring and Σ ⊆ M(R). Let S be a ring (not necessarily graded) and f : R → S be a Σ-inverting ring homomorphism. For γ ∈ Γ, we define the homogeneous rational closure of degree γ as the set (Q f (Σ)) γ consisting of all x ∈ S such that there exist α, β ∈ Γ n and i and x is the (j, i)-entry of (A f ) −1 (for some positive integer n and i, j ∈ {1, . . . , n}). The homogeneous rational closure is the set The graded rational closure, denoted by R f (Σ), is the additive subgroup of S generated by Q f (Σ).
When the set Σ is gr-lower semimultiplicative, the graded rational closure R f (Σ) is a subring of S as the following results show. Lemma 3.3. Let R be a Γ-graded ring and Σ be a gr-lower semimultiplicative subset of M(R). Let S be a ring and f : R → S be a Σ-inverting ring homomorphism. Fix γ ∈ Γ. For x ∈ S, the following conditions are equivalent. (1) x ∈ (Q f (Σ)) γ .
(2) There exist α, β ∈ Γ n and A ∈ Σ n [α][β] such that α i = e, β j = γ and x is the such that such that x is the (j, i)-entry of (A f ) −1 and there exist i, j such that γ = (α i β −1 j ) −1 = β j α −1 i . Then A can be regarded as a matrix in A ∈ Σ n [α · α −1 i ][β · α −1 i ] and thus (2) follows. (4)⇒(5) Let A ∈ Σ, i, j, a and u be as in (4). Suppose that A f u = a f with Now (5) follows from the following equality [e] and u ∈ M n×1 (S) with u ∞ = x. Then the equality A f u = a f is equivalent to the equality Theorem 3.4. Let R be a Γ-graded ring and Σ be a gr-lower semimultiplicative subset of M(R). Let S be a ring and f : R → S a Σ-inverting ring homomorphism. Then [e] and u ∈ M n×1 (S) such that β ∞ = γ, u ∞ = x and and v ∈ M n ′ ×1 (S) such that β ′ ∞ = γ, v ∞ = y and

Then the matrix
[e] and we have the following equality [e] and we have the equality Hence xy ∈ (Q f (Σ)) γδ . From (1)-(3), it is easy to show that R f (Σ) is a Γ-almost graded ring and a subring of S.
(4) Let g, h : R f (Σ) → T be ring homomorphisms. If x ∈ (Q f (Σ)) γ , then x is an entry of a square matrix B which is the inverse of A f for some A ∈ Σ. From (5) Now suppose that S is a Γ-graded ring and f : R → S is homomorphism of Γ-graded rings. Let By (1)-(3), it is easy to prove that R f (Σ) is a graded subring of S whose set of homogeneous elements equals Q f (Σ). Let  (3) If x = 0, then the matrix (A 0 A • ) f is not full over S. Furthermore, if S is a Γ-graded ring and f : R → S is a homomorphism of graded rings, then matrix is not gr-full over S.
Proof. First note the equality (1) Suppose that x is invertible in S. Then Thus xz = 1, yI = 0 and yu • + zx = 1. Therefore x is invertible in S.
(3) Suppose that x = 0. Then (3.1) can be expressed as Given A and x as in Lemma 3.5, we say that Thus, x is invertible in S if and only if its numerator is invertible in M n (S).
Theorem 3.6. Let R be a Γ-graded ring. Let S be a ring and f : R → S be a ring homomorphism. Set Moreover, if S is a Γ-almost graded division ring and f : R → S is a homomorphism of Γ-almost graded rings, then R f (Σ) is a Γ-almost graded division subring of S.
. This can also be expressed as By Lemma 3.3(7), and observing the equality The second part follows because, by Theorem 3.4, R f (Σ) is a Γ-almost graded subring of S which, by the foregoing, is closed under inverses of homogeneous elements.
Corollary 3.7. Let R be a Γ-graded ring, K be a Γ-graded division ring and f : R → K be a homomorphism of Γ-graded rings. If We end this section with an interesting result, but that will not be used in later sections. We show that two elements (and by induction any finite number of elements) can be brought to a common denominator. If x ∈ (Q f (Σ)) γ and y ∈ (Q f (Σ)) δ for some γ, δ ∈ Γ, then they can be brought to a common denominator.

The category of graded R-division rings and gr-specializations
This section is an adaptation of [7, Section 7.2] to the graded situation.

Throughout this section, let Γ be a group
Now we give some important properties of R Σ .
Proof. First we construct a free ring Z X where X is constructed as follows. For each γ ∈ Γ and r ∈ R γ , consider a symbol and consider a matrix A * whose entries are symbols A * = (a * ij ). Then let X be the disjoint union . Now we turn Z X into a Γ-graded ring by giving degrees to the elements of X.
. Let I be the ideal of Z X generated by the homogeneous elements of any of the following forms Set R Σ = Z X /I and λ : R → R Σ be the homomorphism of Γ-graded rings determined by λ(r) = x γ r for each r ∈ R γ , γ ∈ Γ. Since I is a graded ideal of Z X , then R Σ is a Γ-graded ring and λ is a homomorphism of graded rings. Suppose Note that I ⊆ ker F , and let F : R Σ → S be the induced homomorphism. Hence F λ = f , as desired. To prove the uniqueness and the fact that λ : R → R Σ is a ring epimorphism, notice that from F λ = f , we obtain that F (x γ r ) = f (r), and now the same argument of Theorem 3.4(4) shows that F a * ij = b ij . Now we proceed to show (4). If S is a Γ-graded ring and f : . Hence F ′ and F are homomorphisms of Γ-graded rings. Now R f (Σ) is a subring of S generated by Im f and the entries of the inverses of the matrices in Σ f , and that is exactly the image of F . Now our aim is to show that if (K, ϕ) is a Γ-graded epic R-field, then ϕ : R → K is in fact an epimorphism of (Γ-graded) rings. For the sake of completion, we preferred to give the proof of the following lemma, but this could be shown as a direct consequence of [7, Proposition 7.2.1] and the fact that if f : R → S is a homomorphism of Γ-graded rings that is an epimorphism in the category of Γgraded rings, then it is an epimorphism in the category of rings. The proof of this fact is as follows, if g 1 , g 2 : S → T are homomorphisms of rings such that g 1 f = g 2 f , there exist homomorphisms of Γ-graded rings g 1 : S → Im g 1 f , g 2 : S → Im g 2 f and homomorphism of rings π : Im g 1 f → T such that g 1 f = g 2 f and g 1 = π g 1 , g 2 = π g 2 . Since f is an epimorphism of Γ-graded rings, then g 1 = g 2 . Thus g 1 = g 2 . (1) f is an epimorphism of Γ-graded rings.
It can be endowed with a structure of Γ-graded ring via the multiplication (x, u)(y, v) = (xy, xv + uy). Notice that if (x, u) ∈ M γ and (y, v) ∈ M δ , then x, u have degree γ and y, v have degree δ. Hence xy and xv + uy have degree γδ.
(2) ⇒ (1) Let g, h : S → T be homomorphisms of Γ-graded rings such that gf = hf . Then there exists a well defined map F : Thus g = h, as desired.
(2) ⇒ (3) First note that f is a homomorphism of Γ-graded S-bimodules. Clearly f is surjective. Now, since f ( i x i ⊗ y i ) = i x i y i , injectivity follows from the fact that and f is an isomorphism, the result follows. Proof. Suppose that f : R → K is an epimorphism of Γ-graded rings. Consider the graded division subring DC(f ) of K. Let B be a set of homogeneous elements of K that is a basis of K as a right DC(f )-module. Then we have the following isomorphisms of graded right DC(f )-modules and therefore an epimorphism of Γ-graded rings.
The following assertions hold true.
If m is the maximal graded ideal of R Σ , then R Σ /m is a Γ-graded epic R-division ring satisfying the following statements.
(i) There exists a surjective homomorphism of Γ-almost graded rings F : R Σ /m → K such that the following diagram is commutative Proof.
(2) Let λ : R → R Σ be the canonical homomorphism. Hence there exists a unique homomorphism of Γ-almost graded R-rings F : Then F (x) = 0 and then F (x) ∈ K γ is invertible in K. By Proposition 4.1(4), Applying F to the entries of the matrices involved we obtain (i) and (ii) follow, respectively, because m ⊆ ker F and m = ker F if K is a Γ-graded division ring. Now we proceed to define the category of graded epic R-division rings and grspecializations. Let R be a Γ-graded ring. Suppose that (K, ϕ), (L, ψ) are Γ-graded epic R-division rings and set If there exists a homomorphism of Γ-graded R-rings Φ : R Σ → K, we define the core of L in K as C L (K) = Φ(R Σ ). We remark that, if it exists, it is unique and observe that, by Proposition 4.1(4), C L (K) = R ϕ (Σ). By Theorem 4.4(2)(a), R Σ is a Γ-graded local ring. Therefore C L (K) is a Γ-graded local local subring of K that contains R. Moreover, the natural homomorphism of Γ-graded R-rings Ψ : Note that K f is a graded local subring of K because any homogeneous element not in the graded ideal ker f is invertible. Hence K f / ker f is a Γ-graded R-division ring contained in L. This implies that f is a surjective homomorphism of Γ-graded R-rings and that K f / ker f ∼ = L is a Γ-graded epic R-division ring. For each A ∈ Σ, consider A ϕ which belong to M(K). Since K f is a Γ-graded local R-ring whose residue graded division ring is L, we get that A ϕ is invertible in K f . Thus there exists a unique homomorphism of graded R-rings Φ : Thus C L (K) is contained in the domain of any subhomomorphism from K to L, it is a Γ-graded local R-subring of K, the restriction of any subhomomorphism to C L (K) is a subhomomorphism and and all such restrictions coincide in C L (K), because of the commutativity of (4.1). Now we give another description of C L (K). Let f : K f → L be a gr-subhomomorphism between the Γ-graded epic R-fields (K, ϕ), (L, ψ).
is contained in the domain of any gr-subhomomorphism, we get that C L (K) = C L (K). Roughly speaking, this equality means that any rational homogeneous expression obtained from the elements of (the image of) R in L makes sense in K and the elements obtained with those rational expressions from the elements of (the image of) R in K form C L (K).
Because if there exist gr-subhomomorphisms between the Γ-graded epic R-division rings (K, ϕ) and (L, ψ), then they all coincide in the core, we make the following definition. A gr-specialization is the unique homomorphism of Γ-graded R-rings f : Suppose that (K, ϕ), (L, ψ) and (M, φ) are Γ-graded epic R-division rings. If f : K f → L and g : L g → M are gr-subhomomorphisms, then the restriction gf : P = f −1 (L g ) → M is a gr-subhomomorphism which will be called the composition gr-subhomomorphism of f and g. Indeed, suppose that z ∈ h(P ) \ ker(gf ).
Since g(f (z)) = 0, then f (z) −1 ∈ L g . As f (z) = 0, and thus z −1 ∈ K f , then z −1 ∈ P . We define the composition of the corresponding gr-specializations, as the gr-specialization corresponding to the composition gr-subhomomorphism of f and g. In other words, the unique homomorphism of Γ-graded R-rings C M (K) → M . It follows that the composition of gr-specializations is associative.
Note that the only subhomomorphism from the Γ-graded epic R-division ring (K, ϕ) to (K, ϕ) is the identity map on K. Therefore C K (K) = K and the corresponding specialization is the identity map.
We define the category E R as the category whose objects are the Γ-graded epic R-division rings and whose morphisms are the gr-specializations. We remark that there is at most one morphism between two objects in this category and that isomorphisms correspond to isomorphisms of Γ-graded R-rings. Indeed, if the composition of two gr-specializations f and g is the identity gr-specialization, then they have to be isomorphisms of Γ-graded R-rings.
An initial object (K, ϕ) in the category E R is a universal Γ-graded epic R-division ring. In other words, there exists a gr-specialization from (K, ϕ) to any other Γgraded epic R-division ring (L, ψ). If moreover, ϕ : R → K is injective, we say that this initial object is a universal Γ-graded epic R-division ring of fractions of R.
Now we give the following important result.
The following statements are equivalent.
Then by what has been explained above, Φ 2 factors through C K2 (K 1 ), and gives the desired specialization.
Now suppose that there exist gr-specializations f : C K2 (K 1 ) → K 2 and g : C K1 (K 2 ) → K 1 . Then the composition gf gives a gr-specialization from K 1 in itself. Thus it has to be the identity. Similarly the composition f g gives a gr-specialization from K 2 in itself. Hence, f is an isomorphism in the category E R of Γ-graded epic R-division rings. Therefore, f is an isomorphism of graded R-rings.
Proof. Suppose there exists a gr-specialization from the Γ-graded epic R-division ring (K, ϕ) to (R Ω , λ). By Theorem 4.5(3), then there exists a (unique) homomor- Now, since R Ω and K are Γ-graded epic R-division rings, the image of R Ω must be K and therefore they are isomorphic as Γ-graded R-rings.
Proof. Consider the canonical homomorphism λ : R → R Σ . Let f : R Σ → L be a homomorphism of Γ-graded rings with L a Γ-graded division ring. Then (DC(f ), f λ) is a Γ-graded epic R-division ring such that the matrices in Σ become invertible. Hence, by Theorem 4.5, Σ ρ consists of invertible matrices in U . Thus there exists a unique homomorphism of Γ-graded rings ψ : Consider a Γ-graded epic R Σ -division ring (K, ϕ). The composition ϕλ : R → K is an epimorphism of Γ-graded rings, because λ and ϕ are. Hence (K, ϕλ) is a Γ-graded epic R-division ring and therefore there exists a specialization from (U, ρ) to (K, ϕλ).
. Then E is the unique Γ-graded epic Tdivision ring, and thus E is a universal Γ-graded epic T -division ring. Notice that E is not a universal localization at matrices in M(R) because the matrices which become invertible in E are already invertible in T since E is the Γ-graded residue division ring of T . The ring U = T × F with T as before and F a Γ-graded field has E and F as Γ-graded epic U -division rings, but only F is a universal localization.

Malcolmson's Criterion
Throughout this section, let Γ be a group.
In this section, we show that the natural extension of the results and arguments of the paper by P. Malcolmson [25] work for Γ-graded rings. The main results of this section, and the only ones that will be used later, are Theorem 5.1 and Corollary 5.2. The proof of Theorem 5.1 is very technical and most of this long section is devoted to prove it.
In this section, for the ease of exposition, we use the following notation. By the expression A is a homomgeneous matrix, we mean A ∈ M • (R). We will also use the terms homogeneous row, homogeneous column to emphasize that the matrix in question is a row or a column, respectively. If A ∈ M m×n (R)[α][β], but we do not want to make reference to the size of A, we will say A is a homogenousmatrix of distribution (α, β). Also, the sequence αγ will be denoted by αγ for each α ∈ Γ n and γ ∈ Γ.
Corollary 5.2. Let Γ be a group, R be a Γ-graded ring and Σ be a gr-multiplicative subset of M(R) consisting of gr-full matrices. Then R Σ is a nonzero Γ-graded ring.
Proof. It is enough to prove that 1 ∈ R e is not in the kernel of the canonical homomorphism of graded rings λ : R → R Σ . Suppose that 1 ∈ ker λ. Then, by Theorem 5.1, there exist L, M, P, Q ∈ Σ, homogeneous rows J, U and homogeneous , respectively. Since Σ is gr-multiplicative, it is also upper gr-semimultiplicative by Remark 3.1. Thus, the matrix ∈ Σ but it is not gr-full, a contradiction. Therefore, 1 / ∈ ker λ. 5.1. Equivalence relation. Let Γ be a group and R be a Γ-graded ring. Let Σ be a gr-lower semimultiplicative subset of M (R).
The right hand side of (5.1) will also be denoted by       P 11 P 12 P 13 P 14 P 21 P 22 P 23 P 24 P 31 P 32 P 33 P 34 P 41 P 42 P 43 P 44 there is a factorization as a product of homogeneous matrices of any of these forms with L, M, P, Q ∈ Σ and with the corresponding distributions where π 1 = α, π 2 = δ, θ 1 = β and θ 2 = ε, and that we have the factorization  where the factors of the right hand side have distributions (2) Suppose P, U, Q, V have distributions (π, ω), (γ, ω), (ω, θ), (ω, e) where π 1 = α, π 2 = δ, θ 1 = β and θ 2 = ε and we have the factorization:  where the factors of the right hand side have distributions where π 1 = α, π 2 = δ, θ 1 = β and θ 2 = ε and that we have the factorization  where the factors of the right hand side have distribution respectively.
Note that it belongs to (T Σ ) γ .
Finally, if r ∈ R γ , we define µ(r) = (r, 1, 1, e, e) ∈ (T Σ ) γ . Now we prove a series of lemmas that show the compatibility of the operations just defined and the equivalence relation ∼.
Lemma 5.5. The following assertions hold true.
Lemma 5.6. The relation ∼ is compatible with the operations defined on the (T Σ ) γ 's. More precisely, the following assertions hold true.
(2) First note that, by (1), it is enough to prove that . Thus, there exist L, M, P, Q ∈ Σ, homogenous rows J, U , and homogeneous columns W, V as in (5.1). The result follows because the matrix  can be expressed as the product of the homogeneous matrices   B, Y, δ, ε). Thus, there exist L, M, P, Q ∈ Σ, homogenous rows J, U , and homogeneous columns W, V as in (5.1).
In Section 5.2, we proved that the operation + is well defined in (R Σ ) γ for each γ ∈ Γ.
Lemma 5.7. Let γ ∈ Γ. Then (R Σ ) γ is an abelian group with sum defined by Proof. The operation is well defined and commutative by Lemma 5.6(2) and (1). Now we show that the operation is associative. Let as desired.
In Section 5.2, we showed that the product functions ( By the foregoing lemma, it is an additive group. We now prove that it is a Γ-graded ring with the induced product.

Lemma 5.8. R Σ is a Γ-graded ring with the product determined by the rule
Proof. By Lemma 5.6(3), the product is well defined. By Lemma 5.5 (3) and (4), the identity element is [1, 1, 1, e, e]. Now we proceed to show that the product is associative.
which shows that the product is associative. It remains to show that the distributive laws are satisfied. Let The fact that (5.4) equals (5.5) follows because the homogeneous matrix  factorizes as the product of homogeneous matrices  where the factors have distributions respectively.
Proof. By Lemma 5.5(1) and (3), µ is a homomorphism of Γ-graded rings. By E i we will denote the column matrix consisting of 1 as its ith-entry and all the other entries are zero, and by E T i its transpose, the row matrix consisting of 1 as its ith-entry and all other entries are zero. T Let A = (a ij ) ∈ Σ an n × n homogeneous matrix of distribution (α, β). We claim that the n × n matrix B = ([E T i , A, E j , α, β]) ij is the inverse of A µ .

A gr-prime matrix ideal yields a graded division ring, and vice versa
This Section is the adaptation to the graded context of the first part of [7, Section 7.3] and the second part of [7, Section 7.4]. For the proof of the main result Theorem 6.5, instead of using an analog of the first part of [7, Section 7.4], we use Corollary 5.2. Theorem 6.5 could also have been proved via a graded version of [23] that can be found in [17].

Throughout this section, let Γ be a group.
Let R be a Γ-graded ring. If (K, ϕ) is a graded epic R-field, the set {A ∈ M(R) : A ϕ is not invertible over K} will be called the singular kernel of (K, ϕ). Now we show that gr-singular kernels are gr-prime matrix ideals. The aim of this section is to show that gr-singular kernels determine graded epic R-division rings in a similar way as commutative R-fields are determined by prime ideals of R.
Given an n × n matrix A with entries in R, if we write A = (A 1 A 2 . . . A n ) we understand that A 1 , . . . , A n are the columns of A. And if we write A = A1 .

An
we understand that A 1 , . . . , A n are the rows of A. Given two matrices A, B ∈ M(R), we define the diagonal sum of A and B as If they differ at most in the i-th column, then we define the determinantal sum of A and B with respect to the i-th column as Similarly, if they differ at most in the i-th row we define the determinantal sum of A and B with respect to the i-th row as The matrix A∇B, when defined, has the same distribution as A and B.
Note that the operation ⊕ is associative. On the other hand, the operation ∇ is not always defined, and as a consequence it is not associative.
Notice that distributive laws are satisfied. More precisely, if C is another homogeneous matrix, then C ⊕ because, for example, AB and AC (BD, BC) may differ on more than 1 row/column. But in some cases we one can apply the distributive law. Let X ∈ M(R) and suppose that either X is a diagonal matrix, or X is a permutation matrix, then Let Γ be a group and let R be a Γ-graded ring. A subset P of M(R) is a gr-prime matrix ideal if the following conditions are satisfied. (PM1) P contains all the homogeneous matrices that are not gr-full; (PM2) If A, B ∈ P and their determinantal sum (with respect to a row or column) exists, then A∇B ∈ P; (PM3) If A ∈ P, then A ⊕ B ∈ P for all B ∈ M(R); (PM4) For A, B ∈ M(R), A ⊕ B ∈ P implies that A ∈ P or B ∈ P; (PM5) 1 / ∈ P; (PM6) If A ∈ P and E, F are permutation matrices of appropriate size, then EAF ∈ P.
We remark that when Γ = {1}, that is, the ungraded case, (PM6) is a consequence of (PM1)-(PM5) as shown in [7, (g) in p.431]. We have not been able to obtain (PM6) from the others in the general graded case.
Proposition 6.1. Let R be a Γ-graded ring. Let K be a Γ-almost graded division ring and ϕ : R → K be a homomorphism of Γ-almost graded rings. Then is a gr-prime matrix ideal. Therefore, the following assertions hold true.
(1) If (K, ϕ) is a Γ-graded epic R-division ring, then the gr-singular kernel of (K, ϕ) is a Γ-gr-prime matrix ideal. (2) Let N be a normal subgroup of Γ and consider R as a Γ/N -graded ring. Let (K, ϕ) be a Γ/N -graded epic R-division ring. Then Proof. Let K be a Γ-almost graded division ring and ϕ : R → K be a homomorphism of Γ-almost graded rings. First suppose that K = DC(ϕ) and let By Theorem 4.4(2), R Σ is a local ring. If m is the maximal graded ideal of R Σ , there exists a surjective homomorphism of Γ-almost graded rings Φ : R Σ /m → K such that the following diagram is commutative If A ∈ M(R) is not gr-full, then A ϕ is not gr-full. Since K is a Γ-graded division ring, A ϕ is not invertible over K. Thus, (PM1) is satisfied.
Let now A, B ∈ P n [α] [β] such that A∇B is defined. We may suppose that A, B differ on the first column. Hence A = (A 1 C 2 . . . C n ) and B = (B 1 C 2 . . . C n ). Since A ϕ and B ϕ are not invertible over K, the columns of A ϕ and B ϕ are right linearly dependent over K. If the columns C ϕ 2 , . . . , C ϕ n are right linearly dependent over K, then the columns of (A∇B) ϕ are right linearly dependent over K and thus A∇B ∈ P. Hence we can suppose that there exist homogeneous elements a 1 , . . . , a n , b 1 , . . . , b n ∈ K, with a 1 , b 1 = 0, such that A 1 a 1 + C 2 a 2 + · · · + C n a n = 0, B 1 b 1 + C 2 b 2 + · · · + C n b n = 0.
Let A ∈ P and B ∈ M(R), then A ϕ is not invertible over K, but then A ϕ ⊕B ϕ = (A ⊕ B) ϕ is not invertible over K. It implies (PM3). Now suppose that A, B ∈ M(R) are such that A ⊕ B ∈ P. It means that the homogeneous matrix A ϕ ⊕ B ϕ is not invertible over K. It implies that either A ϕ or B ϕ is not invertible. That is, A ∈ P or B ∈ P and (PM4) follows.
Clearly, (PM5) is satisfied. Let A ∈ P and E, F be permutation matrices with entries in R. Notice that E ϕ , F ϕ are permutation matrices with entries in K. Thus, if (EAF ) ϕ = E ϕ A ϕ F ϕ were invertible over K, then A ϕ = (E ϕ ) −1 (EAF ) ϕ (F ϕ ) −1 would be invertible over K, a contradiction. Thus EAF ∈ P and (PM6) is shown.  Proof. (1) By (PM1) and (PM2), if A ∈ P, then C ∈ P. Conversely, suppose that C ∈ P. Clearly A = C∇B ′ where B ′ is obtained from B changing the sign of a row or column. Now A ∈ P because B ′ is not gr-full.
(2) Suppose that β = β 1 * β ′ with and c ∈ R β2β −1 Thus, the right hand side is a determinantal sum of A and (A 2 c A 2 . . . A n ), which is not gr-full. Indeed, it is the product of ( and c 1 0 . . . (3) It follows from (PM6).
(4) We show the first statement, the other can be proved analogously. If we write A = (A 1 A ′ ) and C = (C 1 C ′ ), then The second matrix of the right hand side is a matrix with a submatrix that is a block of zeros of size m × (n + 1). Since m + n + 1 > m + n, that matrix is hollow and therefore not gr-full. By (1), Similarly, one can repeat the argument applied to columns of A ′ and C ′ and so on, to obtain the desired result. (4) implies that Σ is lower gr-semimultiplicative. Finally, (PM6) shows that Σ is gr-multiplicative.
(7) First notice that, by (6) and (PM4), a matrix C ∈ M(R) belongs to P if and only if C ⊕ I ∈ P for the identity matrix I of the same size as C.
We claim that C ∈ P if and only if −C ∈ P. Indeed, and the claim is proved. Then and, by the claim, the result follows.
. Since AA −1 = I / ∈ P, (7) implies that A ⊕ A −1 / ∈ P. Now (PM3) shows that A / ∈ P. (9) By (PM2), if A ∈ P, then C ∈ P. Conversely, suppose that C ∈ P. Clearly A = C∇B ′ where B ′ is obtained from B changing the sign of a row or column. More precisely, B ′ is the product of B by a diagonal matrix D whose diagonal elements are 1 or −1. Now B ⊕ D ∈ P because B ∈ P. Thus B ′ ∈ P by (7). Therefore A ∈ P by (PM2).
The proof of Lemma 6.2 is very similar to the one for the ungraded case, see for example [7, p. 430-431]. The main difference is that we were not able to show [7, (d) p. 430] because not every multiple of a column can be added to another column so that the matrix remains homogeneous. As a consequence the proof of Lemma 6.2(7) is also different.
The following result is well known and can be found, for example, in [12, Proposition 1.1.31]. (2) For a matrix A ∈ Σ n [α ′ * e][β ′ * e], if B, the (n, n)-minor of A, is such that B λ is not invertible over R Σ , then (A − e nn ) λ is invertible over R Σ , where e nn denotes the matrix with 1 in the (n, n) entry and zeros everywhere else.
Proof. Consider the canonical homomorphism of Γ-graded local rings λ : R → R Σ . Suppose that R Σ is a Γ-graded local ring with maximal graded ideal m and canonical homomorphism π : such that its (n, n)-minor B is not invertible over R Σ . It is enough to show that (A − e nn ) π is invertible. Some non-trivial left linear combination (over the graded division ring R/m) with homogeneous coefficients of the rows of B π is zero. If we take the corresponding left linear combination of the first n − 1 rows of A π , we obtain (0, 0, . . . , 0, c) where c is homogeneous and c = 0, because A π is invertible. We now subtract from the last row of A, c −1 times this combination of the other rows and obtain the matrix A − e nn , which is therefore invertible in R Σ /m because it is the product of the matrix corresponding to those elementary operations on A π times A π .
Conversely, suppose now that conditions (1) and (2) are satisfied. By Lemma 6.3, it is enough to prove that (R Σ ) e is a local ring. Let x ∈ (R Σ ) e . By Lemma 3.3(3), there exist α, β ∈ Γ n , A ∈ Σ n [α][β] and u ∈ M n×1 (R Σ ) such that α i = e, β j = e, u j = x and A λ u = e i . Since Σ is gr-multiplicative, we may suppose that A ∈ Σ n [α ′ * e][β ′ * e], u n = x and A λ u = e n . Suppose x is not invertible in R Σ . Equivalently, by Lemma 3.5, the matrix (A λ • e λ n ) is not invertible in R Σ . This implies that the (n, n)-minor of (A λ • e λ n ), which is the (n, n)-minor of A is not invertible in R Σ . Hence, we obtain that 1 − x is invertible in R Σ , as desired. Let R be a graded ring and let P be a gr-prime matrix ideal. The universal localization of R at the set Σ = M(R) \ P will be denoted by R P (instead of R Σ ).
Theorem 6.5. Let Γ be a group and R be a Γ-graded ring. The following assertions hold true.
(1) If P is any gr-prime matrix ideal of R, then the localization R P is a graded local ring. Moreover, its residue class Γ-graded division ring is a Γ-graded epic R-division ring such that its gr-singular kernel equals P. ϕ) is a Γ-graded epic R-division ring, with gr-singular kernel P, then P is a gr-prime matrix ideal and the Γ-graded local ring R P has residue class graded division ring R-isomorphic to K. Proof.
(1) First note that Σ = M(R) \ P is a gr-multiplicative subset of M(R) by Lemma 6.2(5). By Corollary 5.2, since Σ consists of gr-full matrices, we get that R P is a nonzero Γ-graded ring. Consider now an n × n matrix A ∈ Σ n [α ′ * e][β ′ * e] and let B be its (n, n)minor. Suppose that B λ is not invertible over R Σ .. Hence, B belongs to P. Write A = (A ′ A n ) where A n is the last column of A. Then Now note that A = (A ′ A n ) ∈ Σ, and that (A ′ − e n ) ∈ P because B ∈ P and P is gr-lower semimultiplicative. By Lemma 6.2(9), A − e nn / ∈ P, or equivalently A − e nn ∈ Σ. Therefore (A − e nn ) λ is invertible over R Σ . Now Lemma 6.4 implies that R Σ is a Γ-graded local ring. By Theorem 4.4(2)(a), the residue class graded division ring is a graded epic R-division ring. By construction, the singular kernel equals P.
(2) By Proposition 6.1, P is a gr-prime matrix ideal. By (1), R P is a Γ-graded local ring and its residue class graded division ring is a graded epic R-division ring with singular kernel P. Then, by Theorem 4.4(b)(ii), K and the residue class graded division ring of R Σ are isomorphic Γ-graded Rrings.
The following is Theorem 4.5, but expressed in terms of gr-prime matrix ideals.
Corollary 6.6. Let R be a Γ-graded ring, (K i , ϕ i ), i = 1, 2, be Γ-graded epic Rdivision rings with singular kernels P i , respectively. The following statements are equivalent.
(1) There exists a gr-specialization from K 1 to K 2 .
Furthermore, if there exists a gr-specialization from K 1 to K 2 and another grspecialization from K 2 to K 1 , the K 1 and K 2 are isomorphic graded R-rings.
Corollary 6.7. Let R be a Γ-graded ring and (K, ϕ) be a graded epic R-division ring with singular kernel P. Suppose that γ ∈ Γ. Consider the universal localization λ : R → R P and let Φ : R P → K be the homomorphism of Γ-graded rings such that ϕ = Φλ.
(1) Let x ∈ K γ . Then x = 0 if and only if its numerator belongs to P.
(2) Let x ∈ (R P ) γ . Then x ∈ ker Φ if and only if its numerator belongs to P.
Proof. Suppose that (A 0 A • ) is the numerator of x.
(1) By Lemma 3.5(1), x is invertible if and only if (A 0 A • ) ϕ is invertible over K. That is, if and only if (A 0 A • ) belongs to P.
(1) By Lemma 3.5(1), x is invertible if and only if (A 0 A • ) λ is invertible over R P . Since R P is a local ring with residue class graded division ring R-isomorphic to K, x is invertible if and only if (A 0 A • ) Φλ is invertible over K. That is, x ∈ ker Φ if and only if (A 0 A • ) belongs to P.
Corollary 6.8. Let R and R ′ be Γ-graded rings with gr-prime matrix ideals P and P ′ , respectively, with corresponding graded epic R-division rings (K, ϕ) and (K ′ , ϕ ′ ) respectively. Let f : R → R ′ be a homomorphism of Γ-graded rings. The following assertions hold true.
(1) f extends to a gr-specialization if, and only if, P f ⊆ P ′ .
(2) f extends to a homomorphism K → K ′ if, and only if, Proof.
(2) If P f ⊆ P ′ and Σ f ⊆ Σ ′ , then P = P ′′ , and therefore the gr-specialization of (1) is in fact an isomorphism by Corollary 6.6.

Gr-matrix ideals
In this section, the concepts, arguments and proofs are an adaptation of the ones in [7, Section 7.3] to the graded context.

Throughout this section, let Γ be a group.
Let R be a Γ-graded ring. A subset I of M(R) is a gr-matrix pre-ideal if the following conditions are satisfied.
(I1) I contains all the homogeneous matrices that are not gr-full; (I2) If A, B ∈ I and their determinantal sum (with respect to a row or column) exists, then A∇B ∈ I; (I3) If A ∈ I, then A ⊕ B ∈ I for all B ∈ M(R); (I4) If A ∈ I and E, F are permutation matrices of appropriate size, then EAF ∈ I.
If, moreover, we have we call I a gr-matrix ideal.
Clearly, M(R) is a gr-matrix ideal. A proper gr-matrix ideal is a gr-matrix ideal different from M(R).   To prove (7), note that if I n ∈ I, for some n ≥ 1, an application of (I5), shows that 1 × 1 matrix 1 ∈ I. By (I3), any identity matrix I m , m ≥ 1, belongs to I. Again, using (I3), I m ⊕ A ∈ I for any positive integer m and matrix A ∈ M(R). By (5), any A ∈ M(R) belongs to I, as desired.
One could think of defining a gr-prime matrix ideal as a gr-matrix ideal I such that the following two conditions are satisfied.
It is easy to prove that any intersection of gr-matrix (pre-)ideals is again a grmatrix (pre-)ideal. Thus, given a subset S ⊆ M(R), we define the gr-matrix (pre-)ideal generated by S as the intersection of gr-matrix (pre-)ideals I that contain S. That is, S⊆I I. Note that this gr-matrix (pre)-ideal is contained in any gr-matrix (pre-)ideal that contains S. Now we fix some notation that will be used in what follows. Let W ⊆ M(R). We say that a matrix C ∈ M(R) is a determinantal sum of elements of W if there exist A 1 , . . . , A m ∈ W, m ≥ 1, such that A 1 ∇A 2 ∇ . . . ∇A m exists for some choice of parenthesis and equals C.
We will write N to denote the subset of M(R) consisting of the matrices which are not gr-full.
We will denote the set of all identity matrices by I. If X ⊆ M(R), we denote by D(X ) the set of all matrices in M(R) which are of the form E(X ⊕ A)F where X ∈ X , A ∈ M(R) and E, F are permutation matrices of appropriate sizes. We remark that we allow A to be the empty matrix O.
Lemma 7.2. Let R be a Γ-graded ring and A be a gr-matrix pre-ideal. Suppose that Σ ⊂ M(R) satisfies the following two conditions Then the following assertions hold true Proof. (1) Let A ∈ A. By (I3), A ⊕ 1 ∈ A. Since 1 ∈ Σ, A ∈ A/Σ. Hence A ⊆ A/Σ and, by (I1), all non gr-full matrices belong to A. Therefore A/Σ satisfies (I1).
Let A, B ∈ A/Σ be such that A∇B is well defined. There exist P, Q ∈ Σ such that A ⊕ P, B ⊕ Q ∈ A. By (I3), A ⊕ P ⊕ Q and B ⊕ Q ⊕ P belong to A. By (I4), Hence A∇B ∈ A/Σ and A/Σ satisfies (I2).
Let A ∈ A/Σ and E, F be permutation matrices of the same size as A. There exists P ∈ Σ such that A ⊕ P ∈ A. Since E ⊕ I and F ⊕ I are also permutation matrices, (I4) implies that (E ⊕ I)(A ⊕ P )(F ⊕ I) = EAF ⊕ P ∈ A. Hence EAF ∈ A/Σ and (I4) is satisfied.
(3) Clearly I satisfies conditions (i) and (ii). Thus A/I is a gr-matrix ideal that contains A by (1). Let now B be a gr-matrix ideal such that A ⊆ B. If A ∈ A/I, then there exists n ≥ 1 such that A ⊕ I n ∈ A ⊆ B. By applying (I5) repeteadly, we obtain that A ∈ B, as desired.
Lemma 7.3. Let R be a Γ-graded ring and let X ⊆ M(R). Let A(X ) be the subset of M(R) consisting of all the matrices that can be expressed as determinantal sum of elements of N ∪ D(X ). The following assertions hold true.
(1) A(X ) is the gr-matrix pre-ideal generated by X .
(2) A(X )/I is the gr-matrix ideal generated by X . Proof. (1) X ⊆ A(X ) because X = I(X ⊕ O)I for all X ∈ X . By definition of A(X ), every homogeneous matrix that is not gr-full belongs to A(X ). By the same reason, if A, B ∈ A(X ) and A∇B is defined, then A∇B ∈ A(X ).
Let A ∈ A(X ) and B ∈ M(R). That A ⊕ B ∈ A(X ) follows from the following three facts. First, for any U, V ∈ M(R), when defined (U ∇V ) ⊕ M = (U ⊕ M )∇(V ⊕ M ). Second, for X ∈ X and U, M ∈ M(R) and permutation matrices If A ∈ A(X ) and E, F are permutation matrices of appropriate size, then EAF ∈ A(X ). This follows from the following facts. First, if U, V ∈ M(R) and E, F are permutation matrices such that E(A∇B)F is defined, then E(A∇B)F = EAF ∇EBF . Second, for X ∈ X , U ∈ M(R) and permutation matrices E, F, P, Q of appropriate sizes then P is not gr-full, and E, F are permutation matrices of appropriate size, then EU F is not gr-full. Indeed, if U = U 1 U 2 , then EU F = (EU 1 )(U 2 F ).
Therefore, A(X ) is a gr-matrix pre-ideal that contains X . Let now B be a gr-matrix pre-ideal such that X ⊆ B. By (I1), N ⊆ B. By (I3) and (I4), E(X ⊕ A)F ∈ B for all X ∈ X , A ∈ M(R) and permutation matrices E, F of appropriate size. By (I2), A(X ) ⊆ B.
(3) By (2), the gr-matrix ideal generated by X equals A(X )/I. By Lemma 7.2(2), A(X )/I is proper if and only if A(X ) ∩ I = ∅. Proof. Let A ∈ M(R) and suppose it is not gr-full. If A = BC, then A ⊕ Z = (B ⊕ Z)(C ⊕ I) for all Z ∈ Z. Thus A ∈ I Z and (I1) is satisfied.
Let A, B ∈ I Z and suppose that A∇B exists. Then (A∇B)⊕Z = (A⊕Z)∇(B ⊕ Z) for all Z ∈ Z. Since A ⊕ Z, B ⊕ Z ∈ I, then (A∇B) ⊕ Z ∈ I for all Z ∈ Z. Hence A∇B ∈ I Z , and (I2) is satisfied.
Let A ∈ I Z and B ∈ M(R). Since A ⊕ Z ∈ I for all Z ∈ Z and I is a gr-matrix ideal, then A ⊕ Z ⊕ B ∈ I for all Z ∈ Z. By (I4), A ⊕ B ⊕ Z ∈ I for all Z ∈ Z. Therefore A ⊕ B ∈ I Z and (I3) is satisfied.
If A ∈ I Z , Z ∈ Z and E, F are permutation matrices of appropriate size, then EAF ⊕ Z = (E ⊕ I)(A ⊕ Z)(F ⊕ I). It shows that EAF ∈ I Z and (I4) is satisfied.
Suppose now that A ∈ M(R) and that A ⊕ 1 ∈ I Z . Hence A ⊕ 1 ⊕ Z ∈ I for all Z ∈ Z. By (I4), A ⊕ Z ⊕ 1 ∈ I for all Z ∈ Z. Now, by (I5), A ⊕ Z ∈ I for all Z ∈ Z, which shows that A ∈ I Z . Therefore (I5) is satisfied.
Let A 1 , A 2 be two gr-matrix ideals of a Γ-graded ring R. The product of A 1 and A 2 , denoted by A 1 A 2 , is the gr-matrix ideal generated by the set A helpful description of A 1 A 2 is given in the following lemma.
Lemma 7.6. Let R be Γ-graded ring and X 1 , Let A 1 be the gr-matrix ideal generated by X 1 , A 2 be the gr-matrix ideal generated by X 2 and A be the gr-matrix ideal generated by X .
As a consequence, for any A, B ∈ M(R), A B = A ⊕ B , where A denotes the gr-matrix ideal generated by {A}.
Proof. First, A ⊆ A 1 A 2 because X 1 ⊕ X 2 ∈ A 1 A 2 for all X 1 ∈ X 1 , X 2 ∈ X 2 . Now observe that X 1 ⊕ X 2 ∈ X ⊆ A for all X 1 ∈ X 1 , X 2 ∈ X 2 . By (I4), X 2 ⊕ X 1 ∈ X ⊆ A for all X 1 ∈ X 1 , X 2 ∈ X 2 . Hence X 2 is contained in the grmatrix ideal A X1 . Thus, A 2 ⊆ A X1 . It implies that A 2 ⊕ X 1 ∈ A for all A 2 ∈ A 2 and X 1 ∈ X 1 . Again by (I4), X 1 ⊕ A 2 ∈ A for all A 2 ∈ A 2 and X 1 ∈ X 1 . Therefore X 1 is contained in the gr-matrix ideal A A2 . Thus A 1 ⊆ A A2 . This means that A 1 ⊕ A 2 ∈ A for all A 1 ∈ A 1 and A 2 ∈ A 2 . Therefore A 1 A 2 ⊆ A. Now we show that gr-prime matrix ideals behave like graded prime ideals of graded rings.
Proposition 7.7. Let R be a Γ-graded ring. For a proper gr-matrix ideal P, the following are equivalent (1) P is a gr-prime matrix ideal.
Proof. Suppose (1) holds true. Let A 1 , A 2 be gr-matrix ideals such that A 1 P and A 2 P. Hence there exist A 1 ∈ A 1 \ P and A 2 ∈ A 2 \ P. Hence A 1 ⊕ A 2 / ∈ P. It implies that A 1 A 2 P. Therefore (2) holds true.
Let A be a gr-matrix ideal. The radical of A is defined as the set √ A = {A ∈ M(R) : ⊕ r A ∈ A for some positive integer r}.
We say that a proper gr-matrix ideal A is gr-semiprime if √ A = A.
Lemma 7.8. Let R be a Γ-graded ring and let A be a gr-matrix ideal. The following assertions hold true.
(1) √ A is a gr-matrix ideal that contains A. Proof.
(1) If A ∈ A, then, for r = 1, we obtain that A = ⊕ 1 A ∈ A. Hence A ⊆ √ A. In particular, all homogeneous matrices which are not gr-full belong to √ A. Thus √ A satisfies (I1). Let A, B ∈ √ A such that A∇B exists. There exist r, s ≥ 1 such that ⊕ r A, ⊕ s B ∈ A. Set n = r + s + 1. To prove that √ A satisfies (I2), it is enough to show that ⊕ n (A∇B) ∈ A. For that aim, using (A∇B) ⊕ P = (A ⊕ P )∇(B ⊕ P ), one can prove by induction on n that ⊕ n (A∇B) is a determinantal sum of elements of the form where each C i equals A or B. By the choice of n, there are at least r C i 's equal to A or at least s C i 's equal to B. Either case, there exist permutation matrices E, F of appropriate size such that It implies that the elements in (7.1) belong to A by (I3). Now (I2) implies that ⊕ n (A∇B) ∈ A, as desired. Let now A ∈ √ A and B ∈ M(R). There exists r ≥ 1 such that ⊕ r A ∈ A. The equality ⊕ r (A ⊕ B) = E((⊕ r A) ⊕ (⊕ r B))F holds for some permutation matrices Therefore EAF ∈ √ A and √ A satisfies (I4).
It means that ⊕ r A ∈ √ A for some positive integer r. Hence there exists a positive integer s such that ⊕ s (⊕ r A) ∈ A. Thus, ⊕ rs A = ⊕ s (⊕ r A) ∈ A. Therefore A ∈ √ A, as desired. (3) Suppose A is a gr-prime matrix ideal and let A ∈ √ A. Hence ⊕ r A ∈ A. By (PM4), A ∈ A, as desired.
Proposition 7.9. Let R be a Γ-graded ring. Suppose that the nonempty subset Σ of M(R) and the gr-matrix ideal A satisfy the following two conditions.
Then the set W of gr-matrix ideals B such that A ⊆ B and B ∩ Σ = ∅ has maximal elements and each such maximal element is a gr-prime matrix ideal.
Proof. Let (C i ) i∈I be a nonempty chain in W . Set C = i∈I C i . It is not difficult to show that C is a gr-matrix ideal. Then clearly A ⊆ C i ⊆ C and C ∩ Σ = ( i∈I C i ) ∩ Σ = i∈I (C i ∩ Σ) = ∅. By Zorn's lemma, W has maximal elements. Suppose that P is a maximal element of W . Since P ∩ Σ = ∅, P is a proper grmatrix ideal. Let A 1 , A 2 be gr-matrix ideals such that P A 1 , P A 2 . Since P is maximal in W , there exist Corollary 7.10. Let R be a Γ-graded ring. Let A be a proper gr-matrix ideal. Then there exist maximal gr-matrix ideals P with A ⊆ P, and such maximal gr-matrix ideals are gr-prime matrix ideals. In particular, if there are proper gr-matrix ideals, then gr-prime matrix ideals exist.
Proof. By Lemma 7.1(7), no identity matrix belongs to A. Apply now Proposition 7.9 to A and Σ = I.
Proposition 7.11. Let R be a Γ-graded ring. For each proper gr-matrix ideal A, the radical √ A is the intersection of all gr-prime matrix ideals that contain A.
Proof. Let P be a prime matrix ideal such that A ⊆ P. If A ∈ √ A, then ⊕ r A ∈ A ⊆ P for some positive integer r. By (PM4), A ∈ P. Thus √ A ⊆ P.
Notice that such A exists because √ A ⊆ P. If we apply Proposition 7.9 to A and Σ = {⊕ r A : r positive integer }, we obtain a grprime matrix ideal P such that A ⊆ P, P ∩ Σ = ∅. Therefore A does not belong to the intersection of the gr-prime matrix ideals that contain A.

Corollary 7.12. Let R be a Γ-graded ring. A proper gr-matrix ideal is gr-semiprime if and only if it is the intersection of gr-prime matrix ideals.
Let R be a Γ-graded ring. By Corollary 7.4, A(N )/I is the least gr-matrix ideal. We define the gr-matrix nilradical of R as the gr-matrix ideal N = A(N )/I. Proof. (1) is equivalent to (2) by Theorem 4.4(2)(b). One could also argue as follows. By Proposition 6.1, (2) implies the existence of gr-prime matrix ideals, and therefore of Γ-graded epic R-division rings by Theorem 6.5.
If (1) holds, the gr-singular kernel of ϕ is a gr-prime matrix ideal by Theorem 6.5. Thus (3) holds.
Theorem 7.14. Let R be a Γ-graded ring. There exists a universal Γ-graded epic R-division ring if and only if the gr-matrix nilradical is a gr-prime matrix ideal.
Proof. By Corollary 6.6, the existence of a universal Γ-graded epic R-division ring is equivalent to the existence of a least gr-prime matrix ideal P. Hence the least gr-matrix ideal A(N )/I ⊆ P is proper. By Proposition 7.11, N is the intersection of all gr-prime matrix ideals. Hence N = P.
Conversely, if N is a gr-prime matrix ideal, then A(N )/I is proper and, by Proposition 7.11, N is the intersection of all gr-prime matrix ideals. Therefore N is the least gr-prime matrix ideal.
Proposition 7.15. Let R be a Γ-graded ring and let P, Q ∈ M(R). There exists a homomorphism of Γ-graded rings ϕ : R → K to a Γ-graded division ring K such that P ϕ is invertible over K and Q ϕ is not invertible over K if and only if no matrix of the form I ⊕ (⊕ r P ) can be expressed as a determinantal sum of matrices of N ∪ D({Q}).
Proof. The existence of such (K, ϕ) is equivalent to the existence of gr-prime matrix ideals P such that Q ∈ P and P / ∈ P. The existence of such gr-prime matrix ideals is equivalent to the condition P / ∈ Q , where Q denotes the gr-matrix ideal generated by Q. Hence it is equivalent to the condition that no matrix of the form ⊕ r P ∈ Q . By Lemma 7.3 (2), Q is of the form A({Q})/I. Therefore, by Lemmas 7.2 and 7.3, everything is equivalent to the condition that no matrix of the form I ⊕(⊕ r P ) can be expressed as a determinantal sum of matrices of N ∪D({Q}), as desired. Suppose that (1) holds true. Then, for each diagonal matrix A as in (4), A ϕ is invertible. Thus, A / ∈ P, the gr-prime matrix ideal given as the gr-singular kernel of ϕ. In particular, A cannot be expressed as the determinantal sum of matrices in N . Thus (4) holds.
Suppose (4)  Suppose now that (3) holds. If there does not exist a Γ-graded epic R-division ring of fractions, then, by Corollary 10.2, there exists nonzero a ∈ h(R) such that a ϕ is not invertible for every homomorphism of Γ-graded rings ϕ : R → K with K a Γ-graded division ring. Hence the 1 × 1 homogeneous matrix (a) belongs to the intersection of all gr-prime matrix ideals, i.e. (a) ∈ N. Hence ⊕ r (a) ∈ A(N )/I. Thus I s ⊕ (⊕ r (a)) = I s ⊕ aI r can be written as a determinantal sum of matrizes of N . Then, since aI s ⊕ I r ∈ M(R) and it is diagonal, aI r+s = (aI s ⊕ I r )(I s ⊕ aI r ) is a determinantal sum of matrices of N , a contradiction. Therefore (1) holds.

gr-Sylvester rank functions
Throughout this section, let Γ be a group.
The aim of this section is to show that the different definitions of gr-Sylvester rank functions (with values in N) given below are equivalent between them and with the definition of a gr-prime matrix ideal, and thus they uniquely determine homomorphisms to graded division rings. We will adapt the definitions, results and proofs of [24, Sections 1 and 3] and [28, p.94-98] to the graded situation. In defining gr-Sylvester rank functions, the main difference with the ungraded case stems from the the fact that, in the graded case, the same matrix A ∈ M • (R) can define more than one homomorphism between Γ-graded free modules. This is reflected in properties (MatRF4), (ModRF4) and (MapRF5) below.
We begin this section providing the different definitions of gr-Sylvester rank functions for a Γ-graded ring (with values in N), together with some of its basic properties. Let (R) such that A has distribution (α, β), B has distribution (δ, ε) and C has distribution (α, ε) for some finite sequences α, β, δ, ε of elements of Γ .
Let r 1 , r 2 be two gr-Sylvester matrix rank functions for R. We say that r 1 ≤ r 2 if r 1 (A) ≤ r 2 (A) for all A ∈ M • (R). In this way, there is defined a partial order in the set of gr-Sylvester matrix rank functions for R.
The following lemma describes some useful properties of gr-Sylvester matrix rank functions.    (7) follows from Let R be a Γ-graded ring. We will denote the forgetful functor from the category of finitely presented Γ-graded R-modules to the category of finitely presented Rmodules by F . The following easy but important remarks are in order. Let R be a Γ-graded ring. A gr-Sylvester map rank function for R is a function ρ on the class of all homomorphisms of Γ-graded (right) R-modules between finitely generated Γ-graded projective R-modules with values on N such that (MapRF1) ρ(1 R ) = 1, where 1 R denotes the identity map on R.
, then ρ(f ) = ρ(f ′ ). Let ρ 1 , ρ 2 be two gr-Sylvester map rank functions for R. We say that ρ 1 ≤ ρ 2 if ρ 1 (f ) ≤ ρ 2 (f ) for all homomorphism f of Γ-graded modules between finitely generated Γ-graded projective modules. In this way, there is defined a partial order in the set of gr-Sylvester map rank functions for R.
The proof of the following remarks can be proved very much as in Lemma 8.1. (4) r(f ) = r(gf ) = r(f h) for all isomorphisms of Γ-graded R-modules between finitely generated Γ-graded projective R-modules g : Q → Q ′ , h : P ′ → P .
for all homomorphisms of Γ-graded R-modules g : P → Q ′ and h : P ′ → Q with P ′ , Q ′ being finitely generated Γ-graded projective R-modules.
(2) We denote by 0 MN the zero homomorphism M → N between the Γ-graded R-modules M and N . From the equalities Hence ρ(0 P Q ) = 0.
(7) follows from Let R be a Γ-graded ring and let (K, ϕ) be a Γ-graded epic R-division ring. One can induce a gr-rank function r ϕ for R defining r ϕ (A) = rank(A ϕ ) for all A ∈ M • (R). Note that non-isomorphic Γ-graded epic R-division rings induce different gr-rank functions because the gr-singular kernels do not coincide, by Theorem 6.5. The aim of this section is to show that there are no other gr-rank functions for R. (2) If the result A ′ of eliminating any of the columns (rows) of A is such that r(A ′ ) < r(A), then A has exactly n columns (rows).
where we have used Lemma 8.1(4) on the second equality. When the column we eliminate is not the last one, the result follows by Lemma 8.1 (4). When A ′ is obtained by eliminating some row, the result can be proved analogously.
(2) We claim that A ′ obtained as the result of eliminating m of the columns of A satisfies r(A ′ ) = n − m. We prove the claim by induction on m.
If m = 1, the result follows from (1). Suppose that the claim holds true for m ≤ k − 1. Let A ′ be the result of eliminating any k columns and a 1 , a 2 be two different columns of A from among those eliminated in obtaining A ′ . By induction hypothesis, where we have used Lemma 8.1(4) in all but the last equality. Thus r(A ′ ) ≤ n − k. On the other hand, applying (1) several times, we get that n − k ≤ r(A ′ ), and the claim is proved.
Since r(A) ≥ 0 for all A ∈ M • (R), the claim proves the result. When we eliminate rows instead of columns, the result follows analogously.
(3) By (2), we can eliminate rows and/or columns of A until we reach a square submatrix of A of size exactly n. Now we are ready to prove the main result of this subsection. Proof. Theorem 6.5 proves that the two ways of describing the correspondence P → r P are equivalent. By the comment at the beginning of Section 8.1, r P is a gr-rank function for R. Moreover, if P ⊆ Q are gr-prime matrix ideals, then r P ≥ r Q because there are more possible square submatrices which are not in P.
(a) Let now r be a gr-Sylvester matrix rank function for R. Let P r be defined as (b) of the statement of the theorem. We have to show that P r is a gr-prime matrix ideal.
Let A ∈ P r , B ∈ M(R). Since r(A) < size of A, then r(A ⊕ B) = r(A) + r(B) < size of A ⊕ B. Thus (PM3) follows.
Let A ∈ M(R) and suppose that A⊕1 ∈ P r . It means that r(A)+1 = r(A⊕1) < 1 + size of A. Hence r(A) < size of A. Therefore A ∈ P r and (PM5) is satisfied.
By Lemma 8.1(4), (PM6) follows. It remains to show (PM2). Let A, A ′ ∈ P r such that A∇A ′ exists with respect to the last column. Suppose that A = (B c), A ′ = (B c ′ ). Then A∇A ′ = (B c + c ′ ). We claim that r(A∇A ′ ) ≤ max{r(A), r(A ′ )}. This claim implies that A∇A ′ ∈ P r . Then the case of the determinantal sum with respect to any other column follows from Lemma 8.1 (4) and the claim. Now we prove the claim.
If r(A) ≤ r(B), then r(A) = r(B) by Lemma 8.1(7). By (8.1), Interchanging the roles of A and A ′ , we get that if r(A ′ ) ≤ r(B), then r(A) ≥ r(A∇A ′ ) and that if r(A ′ ) > r(B), then r(A ′ ) ≥ r(A∇A ′ ). Thus, the claim is proved. It remains to show that the maps P → r P and r → P r are inverse one of the other.
If P is a gr-prime matrix ideal, the gr-prime matrix ideal that corresponds to r P is the set of matrices A ∈ M(R) such that r P (A) < size of A. That is, the set of matrices A ∈ M(R) whose largest square submatrix that is not in P is less than the size of A. In other words, the matrizes A ∈ P. Therefore r P → P. On the other hand, let now r be a gr-rank function for R. Let r Pr be the associated gr-rank function associated to P r . If A ∈ M • (R), then r Pr (A) equals the size of a largest square submatrix of A which is not in P r . That is, the size of a largest square submatrix B of A such that r(B) = size of B. Hence we have to show that r(A) = n if and only if n is the size of a largest square submatrix of A such that r(B) = n. But this now follows from Lemma 8.4(3).

8.2.
Equivalence between gr-Sylvester rank functions. The following result can be proved in exactly the same way as in [24, Lemma 2] where the ungraded case is shown. Lemma 8.6. Let R be a Γ-graded ring. If 0 → K → Q → M → 0 and 0 → K ′ → Q ′ → M → 0 are exact sequences of Γ-graded R-modules with Q and Q ′ Γ-graded projective R-modules and with K ⊆ Q and K ′ ⊆ Q ′ , then there is an automorphism The next result was first stated in [28, p. 97] for the ungraded case. Our proof follows the one of [24, Theorem 4]. We would like to remark that the fact that N is the set of values of Sylvester rank functions is not used in the proof.
(b) Suppose that ρ is a gr-Sylvester map rank function. If M is a finitely presented Γ-graded R-module and f : P → Q is a homomorphism of Γ-graded R-modules with P, Q finitely generated Γ-graded projective R-modules such that coker f = M , then we define d ρ (M ) = ρ(1 Q ) − ρ(f ). We must show that d is well defined and satisfies (ModRF1)-(ModRF4).
We begin showing that d ρ is well defined. Suppose that P f → Q → M → 0 and P ′ f ′ → Q ′ → M → 0 are two graded presentations with P, P ′ , Q, Q ′ finitely generated Γ-graded R-modules. By Lemma 8.6, there exists an automorphism h of Γ-graded R-modules of Q ⊕ Q ′ which maps Im f ⊕ Q ′ onto Q ⊕ Im f ′ . Since Q ⊕ Q ′ is a Γ-projective R-module, we obtain the following diagram of homomorphisms of Γ-graded R-modules Hence we obtain h → Q 2 → M 2 → 0 be exact sequences of homomorphisms of Γ-graded R-modules with P 1 , P 2 , Q 1 , Q 2 finitely generated Γ-graded projective R-modules. → M 3 → 0 be exact sequences of homomorphisms of Γ-graded R-modules with P 1 , Q 1 , P 3 , Q 3 finitely generated Γ-graded projective R-modules. By diagram chasing, it is easy to obtain the following commutative diagram, with exact rows and columns, of homomorphisms of Γ-graded R-modules where ι and π are the natural inclusion and projection, respectively 0 0 ker g π | ker g / / Since ker g is a finitely generated Γ-graded R-module, there exist a finitely generated Γ-graded projective R-module and a surjective homomorphism of Γ-graded R-modules P f → ker g. In this way, we obtain a commutative diagram, with exact rows and columns, of homomorphisms of Γ-graded R-modules P f P πf =f ′ , and properties (MapRF2), (MapRF3) on the first inequality, and Lemma 8.3(6) on the second inequality. Therefore (ModRF3) is satisfied. (ModRF4) follows easily. Indeed, let f : The next result was given in [24, Theorem 4] in the ungraded context.
Proof. First we show that the correspondence is an anti-isomorphism of partially ordered sets. Let r 1 ≤ r 2 be two gr-Sylvester matrix rank functions. Let M be a finitely presented Γ-graded R-module and suppose that R n (β) Secondly, we show that the correspondences are one inverse of the other. Let d be a gr-Sylvester module rank function. Let M be a finitely presented Γ-graded R-module. Then, given a graded presentation of M , R n (α) ). The fact that r d is well defined follows from (ModRF4). That is, the matrix A may define different homomorphisms of Γ-graded modules, but the value of r d (A) is the same. The proof that r d satisfies (MatRF1)-(MatRF4) follows in the same way as the proof that ρ d satisfies (MapRF1)-(MapRF5) in Theorem 8.7.
(b) Suppose now that r is a gr-Sylvester matrix rank function for R. Let M be a finitely presented Γ-graded R-module with presentation R n (β) [β], we define d r (M ) = m − r(A). One can show that d r is well defined and satisfies (ModRF1)-(ModRF4) in the same way that one proves that d ρ is well defined and satisfies (ModRF1)-(ModRF4). There is another way to prove that d r is well defined and satisfies (ModRF1)-(ModRF4). By Theorem 8.5, let (K, ϕ) be the corresponding Γ-graded epic R-division ring with r. Then, for a finitely presented Γ-graded R-module M , the K-module M ⊗ R K is a Γ-graded free K-module. One can define d(M ) = dim K (M ⊗ R K). It is not difficult to show that d satisfies (ModRF1)-(ModRF4). Now let M be a finitely presented Γ-graded R-module with presentation R n (β) . Then, by Theorem 8.5, dim(M ⊗ R K) equals m minus the number of columns of A ϕ which are right linearly independent over K and that is exactly m − r(A) = d ρ (M ).
It is worth noting the following corollary. It is just a re-writing of parts of Theorems 4.5, 8.7, 8.8 and Corollary 6.6. Let R = γ∈Γ R γ be a Γ-graded ring. In the foregoing, we gave a correspondence from the set of Γ/Ω-graded epic R-division rings to the set of Γ-graded epic R-division rings. We proceed to give a more down to earth description of such correspondence. Recall that R can be regarded as a Γ/Ω-graded ring making R = ∆∈Γ/Ω R α where R α = γ∈α R γ for each α ∈ Γ/Ω.
Let A = (a ij ) ∈ M n (R)[δ][ε]. We claim that A ϕ is invertible in E if and only if A ψ is invertible in D. Indeed, let α i , β j ∈ Γ/Ω be such that δ i ∈ α i , ε j ∈ β j .
. Hence, let P ∈ Spec Γ/Ω (R). If (E, ϕ) is the Γ/Ω-graded epic R-division ring associated to P, then the Γ-graded epic R-division ring associated to P ∩ M Γ (R) is determined by the Γ-graded division ring ψ : R → D. That is, the Γ-graded epic R-division ring ψ : R → D ′ where D ′ the graded division ring generated by Im ψ.
We would like to remark that D(Γ), the division subring of D((Γ; <)) generated by D, does not depend on the order < of Γ by [13] or [8]. Hence, since D(Ω) is just DC(φ Ω ), then D(Ω) does not depend on the order < of Γ.
We end this section with a concrete application of the results in this section. Let K be a field, X be a nonempty set and K X be the free K-algebra on X. It is well known that K X has a universal division ring of fractions [7, Section 7.5]. Let now Γ be a group and X → Γ, x →x, be a map. Then K X = γ∈Γ K X γ is a Γ-graded ring where K X γ is the K-vector space spanned by the monomials x 1 x 2 . . . x r such thatx 1x2 · · ·x r = γ. If (Γ, <) is an ordered group, then K X has a Γ-graded universal division ring of fractions by the foregoing example and Theorem 9.1(3),(4).

Inverse limits and ultraproducts in the category of graded epic R-division rings
For details on filters and ultrafilters we refer the reader to [4]. Let I be a nonempty set. A filter on I is a set F of subsets of I which has the following properties (F1) Every subset of I that contains a set of F belongs to F. (F2) Every finite intersection of sets of F belongs to F. (F3) The empty set is not in F.
The set of filters on I is partially ordered by inclusion. An ultrafilter on I is a maximal filter. By [4, Theorem 1, p.60], each filter is contained in an ultrafilter. An ultrafilter U on I has the following property: if J, K are subsets of I such that J ∪ K = I, then either J ∈ U or K ∈ U.
The concrete ultrafilters we will be dealing with are constructed as follows. Let (I, ≤) be a directed preordered set. For each i ∈ I, the set S(i) = {j ∈ I : i ≤ j} is called a section of I relative to i. The set S consisting of all sections relative to elements of I is a filter base and there exists a filter containing S [4, Proposition 2, p.59]. Therefore there exists an ultrafilter of I containing S.
Let Γ be a group. Let I be a set and U be an ultrafilter on I. For each i ∈ I, let R i = i∈I R iγ be a Γ-graded ring. We proceed to define the graded ultraproduct of the family {R i } i∈I following [14]. Consider the ring P = i∈I R i and consider the following subset S of P S = γ∈Γ i∈I R iγ .
Note that S is a subring of P which is Γ-graded with S γ = i∈I R iγ . For each γ ∈ Γ, if x = (x iγ ) i∈I ∈ S γ , let z(x) = {i ∈ I : x iγ = 0}. The set Z γ = {x ∈ S γ : z(x) ∈ U} is an additive subgroup of S γ . Moreover, if y ∈ S δ and x ∈ Z γ , then yx ∈ Z δγ and xy ∈ Z γδ . Therefore Z = γ∈Γ Z γ is a graded ideal of S. Then the Γ-graded ring U = S/Z is called the graded ultraproduct of the family of Γ-graded rings {R i } i∈I .
A homogeneous element element x ∈ U γ is the class of an element (x i ) i∈I ∈ S γ , where each x i ∈ R iγ . We will write x = [(x i ) i∈I ] U . Observe that if x = [(x i ) i∈I ] U and y = [(y i ) i∈I ] U , then x = y if and only if the set {i ∈ I : x i = y i } ∈ U.
Suppose that (R i , ϕ i ) is a Γ-graded R-ring for each i ∈ I. Hence ϕ i : R → R i is a homomorphism of Γ-graded rings. Then there exists a unique homomorphism of rings ϕ ′ : R → i∈I R i such that π i ϕ ′ = ϕ i for each i ∈ I. Observe that Im ϕ ′ ⊆ S. Composing with the natural homomorphism S → S/Z = U , we obtain a homomorphism of Γ-graded rings ϕ : R → U . Hence U is a Γ-graded R-ring in a natural way. This fact and the following lemma will be very useful in this section.
Lemma 10.1. Let Γ be a group. Let I be a nonempty set and U be an ultrafilter on I.
(1) If R i is a Γ-graded division ring for each i ∈ I, then the ultraproduct U of the family {R i } i∈I is a Γ-graded division ring. (2) If R i is a Γ-graded local ring with graded maximal ideal m i for each i ∈ I, then the ultraproduct U of the family {R i } i∈I is a Γ-graded local ring with residue Γ-graded division ring V , the ultraproduct of the family of Γ-graded division rings {R i /m i } i∈I .
Proof. (1) Let x ∈ U γ . Then x = [(x i ) i∈I ] U for some x i ∈ R iγ . If x is nonzero, then J = {i ∈ I : x i = 0} ∈ U. For each i ∈ I, define Notice that x i ′ ∈ R iγ −1 for each i ∈ I. Then x ′ = [(x ′ i ) i∈I ] U ∈ U γ −1 and xx ′ = x ′ x = 1, as desired.
(2) A homogeneous element x = [(x i ) i∈I ] U is invertible in U if and only if the set {i ∈ I : x i is invertible in U } ∈ U if and only if {i ∈ I : x i / ∈ m i } ∈ U. Therefore, the ideal m generated by the homogeneous noninvertible elements, that is, the set {[(x i ) i∈I ] U ∈ h(U ) : {i ∈ I : x i ∈ m i } ∈ U}, is a proper ideal of U . Hence U is a graded local ring.
It is not difficult to prove that the projections R i → R i /m i , a → a, induce a surjective homomorphism of Γ-graded rings U → V , [(x i ) i∈I ] U → [(x i ) i∈I ] U . Note that a homogeneous element [(x i ) i∈I ] U is in the kernel if and only if it belongs to m.
The following corollary was used in Section 7 Corollary 10.2. Let R be a Γ-graded domain. Suppose that, for each a ∈ h(R) \ {0}, there exists a homomorphism of Γ-graded rings ϕ a : R → K a , where K a is a Γ-graded division ring such that ϕ a (a) = 0. Then there exists a Γ-graded epic R-division ring of fractions.
Proof. Let I = h(R) \ {0}. For each a ∈ I, let I a = {λ ∈ I : ϕ λ (a) = 0}. Let E = {a 1 , . . . , a n } be a finite subset of I. Then n i=1 I ai = ∅, because ϕ a1···an (a i ) = 0 for each i = 1, . . . , n. Hence the set B = {I a : a ∈ I} is a set of subsets of I such that no finite subset of B has empty intersection. By [4, Proposition 1,p.58], there exists a filter on I containing B. By [4, Theorem 1, p.60], there exists an ultrafilter U on I containing B. By Lemma 10.1(1), the ultraproduct U of the family {K a } a∈I is a Γ-graded division ring and there exists a homomorphism of Γ-graded rings ϕ : R → U , defined by ϕ(x) = [(ϕ a (x)) a∈I ] U . Since the set I x ∈ U, then ϕ(x) = 0 for each x ∈ h(R) \ {0}. Therefore ϕ is injective. Let R be a Γ-graded ring. Consider the category E R of Γ-graded epic R-divison rings with specializations as morphisms defined in Section 4.
First we look at how inverse systems are in this category. An inverse system in E R is a pair ((K i , ϕ i ) i∈I , (ψ i,j ) i≥j ) where (I, ≤) is a directed preordered set, (K i , ϕ i ) is a Γ-graded epic R-division ring for each i ∈ I, and ψ i,j is a specialization from (K i , ϕ i ) to (K j , ϕ j ), i ≥ j, such that ψ j,k • ψ i,j = ψ i,k for all i, j, k ∈ I, i ≥ j ≥ k. (10.1) Observe that (10.1) is superfluous because since the specializations already exist, and there is at most one specialization between graded epic R-division rings, the equality in (10.1) holds trivially. Now we look at inverse limits. An inverse limit of the inverse system ((K i , ϕ i ) i∈I , (ψ i,j ) i≥j ) in E R is a pair ((K, ϕ), (ψ i ) i∈I ) where (K, ϕ) is a Γ-graded epic R-division ring and ψ i is a specialization from (K, ϕ) to (K i , ϕ i ) for each i ∈ I such that the following properties are satisfied: (i) ψ i,j • ψ i = ψ j for all i, j ∈ I, i ≥ j.
Again note that (i) and the equality of specializations in (ii) are superfluous.
Theorem 10.3. Let R be a Γ-graded ring. Let ((K i , ϕ i ) i∈I , (ψ i,j ) i≥j ) be an inverse system in E R indexed on the directed nonempty preordered set (I, ≤). Consider an ultrafilter U on I that contains all the sections S(i), i ∈ I, of I. Set The following assertions hold true.
(1) There exists a Γ-graded epic R-division ring (K, ϕ) which is the inverse limit of ((K i , ϕ i ) i∈I , (ψ i,j ) i≥j ). Proof. For each i ∈ I, let P i be the singular kernel of ϕ i : R → K i .
(1) Since there exists a specialization ψ i,j : K i → K j for i, j ∈ I, i ≥ j, then P i ⊆ P j by Corollary 6.6. Thus, the family of prime matrix idelas {P i } i∈I is directed from below. Set P = i∈I P i . It is not difficult to prove that P satisfies (PM1)-(PM3), (PM5) and (PM6) in the definition of gr-prime matrix ideal. To show that P satisfies (PM4), let A, B ∈ M(R) \ P. There exist i, j ∈ I such that A / ∈ P i and B / ∈ P j . Since I is directed, there exists k ∈ I such that i ≤ k, j ≤ k. Thus P k ⊂ P i and P k ⊆ P j , and both A and B do not belong to the gr-prime matrix ideal P k . Hence A ⊕ B / ∈ P k , and therefore A ⊕ B / ∈ P. Let (K, ϕ) be the Γ-graded epic R-division ring corresponding to P. Since P ⊆ P i for all i ∈ I, there exists a unique specialization ψ i : K → K i . Consider now a pair ((L, ϕ ′ ), (ψ ′ i ) i∈I ) where L is a Γ-graded epic R-division ring and ψ ′ i is a grspecialization from (L, ϕ ′ ) to (K i , ϕ i ) for each i ∈ I. Let Q be the singular kernel of ϕ ′ : R → L. Then Q ⊆ P i for each i ∈ I. Hence Q ⊆ P. Therefore, there exists a specialization φ : L → K by Corollary 6.6.
(3) Let U be the graded ultraproduct of the family ((K i , ϕ i ) i∈I of Γ-graded epic R-division rings, and let ϕ : R → U be the canonical homomorphism of Γ-graded rings. Let L be the Γ-graded division subring of U generated by the image of ϕ. Consider the Γ-graded epic R-division ring (L, ϕ).
Let A ∈ i∈I Σ i be such that A ∈ M n (R)[α][β]for some α, β ∈ Γ n . By Theorem 4.5, note that Σ j ⊆ Σ i for all i, j, i ≥ j. Let t ∈ I be such that A ∈ Σ t . In particular U = ∅. If i ∈ U, then A ∈ Σ i , and therefore A ∈ i∈I Σ i . Hence, the singular kernel of (L, ϕ) is P = M(R) \ ( i∈I Σ i ).
(5) First observe that R Σ = lim − → R Σi . Let τ i : R Σi → R Σ and λ i : R → R Σi be the natural homomorphism of Γ-graded rings. By (3), there exists a natural homomorphism of Γ-graded rings ρ : R Σ → U . Since V is Γ-graded local with residue Γ-graded division ring equal to U , the universal property of R Σ , induces a homomorphism of Γ-graded rings ρ : R Σ → V . We must prove that ρ is injective. Suppose that x ∈ ker ρ is homogeneous. Then x is the (u, v)-entry of the inverse of a matrix A ϕ with A ∈ Σ = i∈I Σ i . Let i ∈ I be such that A ∈ Σ i . Note that A ∈ Σ l for all l ∈ I, i ≤ l. Let x l uv be the (u, v)-entry of A λ l , i ≤ l. Then τ i (x l uv ) = x. Note that the proof of (3) shows also (1) in a more elementary way.