CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION

We give some criteria for recognizing local rings that allow us to show that indecomposable AB5∗ modules over commutative rings and couniform modules over noetherian commutative rings have a local endomorphism ring. We also develop some theory on methods to construct modules with a prescribed direct-sum decomposition. As an application we realize an interesting class of commutative monoids as monoids of direct summands of a direct sum of a countable number of copies of a suitable artinian cyclic module, showing that there may appear a rich supply of direct summands that are not a direct sum of artinian modules. An important gadget for proving our realization result is a variation of a method for realizing a given ring as the endomorphism ring of a cyclic (artinian) module due to Armendariz, Fisher


Introduction
One of the fundamental tools to describe the direct sum decompositions of a module is to study the projective modules over its endomorphism ring. A. Dress was the first to state the existence of a category equivalence between the category of modules that are isomorphic to direct summands of M n , for some n and a fixed right module M over a ring R, and the category of finitely generated projective right modules over End R (M ), cf. Proposition 6.1. Therefore, knowing the endomorphism ring of a module M and the behavior of its finitely generated projective modules is equivalent to knowing the behavior of the direct sum decomposition of M n for any n. Hence, in order to construct examples and counterexamples in the setting The author was partially supported by DGI MINECO MTM2011-28992-C02-01, by ERDF UNAB10-4E-378 "A way to build Europe", and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
of direct sum decompositions it is extremely useful to be able to construct modules with a prescribed endomorphism ring.
Armendariz, Fisher and Snider were studying in [1] when every injective/onto endomorphism of a finitely generated module over a PI ring is bijective. They constructed in [1,Example 3.2] an example that was quite interesting for their context, but having a closer look people realized that their idea was giving a method to construct cyclic modules with a prescribed endomorphism ring. The further developments of Armendariz, Fisher and Snider's method have had an impact in the theory of direct sum decomposition of modules in general, and of direct sum decompositions of artinian modules in particular. We explain the pattern of their idea.
Let This formulation is due to Camps and Menal [4], they also realized that if A/S is an artinian right S-module and A A is artinian then (T /I) T is artinian. Camps and Menal used this to give some interesting and non-trivial examples of artinian modules. Later Camps and Facchini [3] developed a more sophisticated statement, which is the one we give in Lemma 6.3, that allowed any finitely generated algebra over a semilocal commutative noetherian ring to be realized as endomorphism ring of a cyclic artinian module, cf. [8]. Since the Krull-Schmidt Theorem fails for finitely generated projective modules over a finitely generated algebra over a semilocal noetherian ring, one could conclude that artinian modules also fail to satisfy the Krull-Schmidt Theorem. This answered in the negative a question posed by Krull in 1932 [19].
In our main Theorem 6.11 we will give one further application of this tool to construct artinian modules with a prescribed endomorphism ring. We show that the category of direct summands of a direct sum of a countable number of copies of a cyclic artinian module can have a rich supply of direct summands that are not a direct sum of artinian modules, cf. Example 7.12.
The paper is divided into six sections. The first two are devoted to giving new classes of indecomposable modules, over commutative rings, with local endomorphism ring. The first class is that of indecomposable AB-5 * modules over a commutative ring (Proposition 2.4) and the second one is the class of couniform modules over a commutative noetherian ring (Corollary 3.4). Our proofs are quite elementary and self-contained so we think that they are also interesting in the known cases they cover. For example, as a further outcome, we get a new proof of the fact that over a commutative ring an indecomposable artinian module has a local endomorphism ring.
Examples of the failure of the Krull-Schmidt Theorem for artinian modules were also constructed by Pimenov and Yakovlev in [25] and by Ringel in [28]. Their strategy was to give explicit equivalences between suitable and well understood categories of modules and categories of artinian modules over triangular matrix rings. In Section 3 we give a general setting for these constructions and we develop further application in Section 4.
After the work of Facchini and Herbera [7] and the subsequent by Wiegand [31], the direct sum behavior of artinian modules and, in general, of finite direct sums of modules with a semilocal endomorphism ring is relatively well understood. The new tool introduced for that are the monoids of isomorphism classes of finitely generated projective modules and the monoids of isomorphism classes of direct summands of a finite number of copies of a given module. Understanding the structure of these monoids is equivalent to understanding the behavior, in direct sums, of the involved modules. We recall this machinery and we explain the specific tools needed for the case of a semilocal ring in Section 5.
Right now a very challenging question is to understand the behavior of infinite direct sums of modules with a semilocal endomorphism ring. In [16], Herbera and Příhoda characterized the monoid of isomorphism classes of countably generated projective modules over a semilocal noetherian ring. In the main result of this paper (Theorem 6.11) we show that these monoids can be realized as the monoid of isomorphism classes of direct summands of an infinite sum of copies of an artinian module. Section 6 is devoted to proving this result, our strategy is to use the variations of Armendariz, Fisher and Snider method to show that the rings constructed in [16] can be also realized as endomorphism rings of suitable cyclic artinian modules. In order to do that and because the rings from [16] are constructed via pullbacks another important tool in the proof of Theorem 6.11 is the characterization of injective modules over pullbacks due to Facchini and Vámos [9].
2. Criteria for recognizing a local ring. An application to AB-5 * modules All our rings are associative with 1, and ring morphism means unital ring morphism.
We recall that a ring R is said to be semilocal if modulo its Jacobson radical J(R) is semisimple artinian.
In this section we give a couple of (easy) criteria for proving that a ring is local just by looking at certain families of commutative subrings.
Our general philosophy to decide whether some classes of modules over commutative rings have local endomorphism ring is to "enlarge the ring". If M is an R-module over a commutative ring then R/Ann R (M ) embeds canonically in End R (M ). In fact M can be viewed as a module over any ring T such that R/Ann R (M ) ⊆ T ⊆ End R (M ), and the endomorphism ring of M as T -module will be a subring of End R (M ). Note that if T is a maximal commutative subring of End R (M ) containing R/Ann R (M ), then the endomorphism ring of M as T -module is the same ring T . Proposition 2.1. Let R ⊆ S be a ring extension, such that R is in the center of S. Then S is local if and only if every maximal commutative subring of S containing R is local.
Proof. Assume that S is local and that T is a maximal commutative subring of S containing R. If t ∈ T then either t or 1 − t is invertible in S. The maximality of T implies that the inverse of an element in T is also an element of T , so either t or 1 − t is invertible in T . This implies that T is local.
Conversely, let s ∈ S. Consider a maximal commutative subring T of S containing s and R. Then 1 − s ∈ T and since, by hypothesis, T is local either s or 1 − s is invertible in T , hence in S.
Let R be a subring of the center of a ring T . Let t ∈ T and consider the ring extension R ⊆ R[t] ⊆ T . Let Σ be the set of all elements in R[t] that are invertible elements in T . Then Σ is a multiplicatively closed subset of R[t], and R[t] Σ can be identified with a subring of T . In next proposition we shall denote this ring by R t . The following result and its proof is a variation of Proposition 2.1.
Proposition 2.2. Let R ⊆ T be a ring extension, such that R is in the center of T . Then T is local if and only R t is local for any t ∈ T .
Let R be a ring. A right R-module M satisfies the AB-5 * property provided that for any inverse system of submodules of M , {M i } i∈I say, and for any submodule N of M the following equality holds true: It is clear from the definition that the AB-5 * property is inherited by submodules and quotients. Also, as it is a lattice property, if R ⊆ T is a ring extension and M T is a T -module that is AB-5 * as an R-module then it is also AB-5 * as a T -module.
Examples of modules satisfying the AB-5 * property are Artinian modules and, in general, modules that are linearly compact with the discrete topology. Uniserial modules are also examples of AB-5 * modules, and a semisimple module is AB-5 * if and only if all its isotypic components have finite length.
It is well known that, over a commutative ring, indecomposable linearly compact modules and uniserial modules have local endomorphism ring. We shall prove that this is true also for indecomposable AB-5 * modules.
A module M is said to be complemented if for each submodule X of M there is a submodule Y , called the (addition) complement of X, minimal with respect to the property Y +X = M . As it was observed by Lemonnier in [20], a trivial application of Zorn's Lemma shows that an AB5 * module M is complemented. Kasch and Mares' proved that a ring is semiperfect if and only if R R is complemented if and only if R R is complemented, hence left or right AB5 * rings are semiperfect. This implies that if M R is an AB-5 * R-module over a commutative ring R and T is some commutative subring of End R (M ) then, for any m ∈ M , T /Ann T (x) is a semiperfect ring because mT ∼ = T /Ann T (x) is an AB5 * T -module. This observation will be the key ingredient in proving that, over a commutative ring, indecomposable AB-5 * modules have local endomorphism ring.
Before proving the result we recall the following facts from [18, Lemma 8]: Remark 2.3. Let R be a commutative ring, and let V be a fixed simple R-module.
For any R-module M we consider the following subset of M Let {V i } i∈I be a family of representatives of the isomorphism classes of simple modules over R, and consider the family R-submodules of M , {M Vi } i∈I . It is clear that {M Vi } i∈I is a family of independent R-submodules. If the module M satisfies that for any m ∈ M , End R (mR) ∼ = R/Ann R (m) is a semiperfect ring (e.g. if M is AB-5 * ), then M = ⊕ i∈I M Vi . Proposition 2.4. Let R be a commutative ring and M R an indecomposable AB-5 * module over R. Then End R (M ) is a local ring.
Proof. We shall prove that any maximal commutative subring of End R (M ) is local, and then the result will follow from Proposition 2.1.
Let T be a maximal commutative subring of End R (M ), and note that M T is AB-5 * . Let {V i } i∈I be a set of representatives of the isomorphism classes of simple modules over T . Then, by Remark 2.3, M = ⊕ i∈I M Vi . As M is indecomposable there exists i ∈ I such that M = M Vi . Let M = Ann T (V i ) and let t ∈ T \ M. Since for any 0 = m ∈ M , T /Ann T (m) ∼ = End T (mT ) is a local ring with maximal ideal M/Ann T (m) then, for any m ∈ M , the endomorphism of mT induced by multiplication by t is bijective. Hence, multiplication by t is a bijective endomorphism of M . As T is a maximal commutative subring of End R (M ), t −1 ∈ T . Therefore M is the unique maximal ideal of T , and T is local.
Theorem 2.5. Let M be an AB-5 * module over a commutative ring R. Then the following statements hold. On the other hand, by the above argument, it also follows that, for i = j ∈ I, Hom R (M i , M j ) = 0. This finishes the proof of (1).
To prove (2) recall that a module has a semiperfect endomorphism ring if and only if it is a finite direct sum of submodules with local endomorphism ring. Hence, by (1), M is a direct sum of modules with local endomorphism ring. Also from (1) it follows that End R (M ) is a product of semiperfect rings.
As a product of semiperfect rings is a ring that is von Neumann regular modulo the Jacobson radical and idempotents can be lifted modulo it, we deduce from [29,Theorem 3] that End R (M ) is an exchange ring. Hence M satisfies the finite exchange property, so it also satisfies the exchange property by [33,Corollary 6].
As linearly compact modules satisfy AB−5 * and have finite Goldie dimension we obtain the following well known corollary of Theorem 2.5. Remark 2.7. Linearly compact modules over a non-necessarily commutative ring may not have a semiperfect endomorphism rings but they have a semilocal endomorphism ring [18]. However if R is a ring with right Morita duality then all linearly compact right R-modules are pure injective, hence their endomorphism ring is also semiperfect.
We do not know whether the endomorphism ring of a linearly compact module over a commutative ring is linearly compact. This question was considered in [10] and it was proved to be true in a number of cases, e.g. for linearly compact modules over commutative noetherian rings.

Couniform modules
A nonzero module M is said to be couniform if the sum of two proper submodule of M is a proper submodule of M . A module that is uniform and couniform is called biuniform.
We recall the following facts about couniform modules, Then: (i) if f and g ∈ I, then f + g ∈ I.
(ii) gf ∈ I if and only if f ∈ I or g ∈ I.
In particular, I is an ideal of End R (M ).
Note that, by Lemma 3.1, if M R is a couniform module and S is a subring of End R (M ), then S ∩ I is a completely prime ideal of S. Proposition 3.2. Let R be a commutative ring, and let M R be a couniform module. Let r be an element of R such that neither multiplication by r nor by 1 − r is a bijective endomorphisms of M .
(i) If multiplication by r is not onto then there exists a biuniform R-module N with essential socle such that the endomorphism ring of N is not local. (ii) If multiplication by r and by 1 − r are onto endomorphisms that are not injective, then there exists a couniform R-module N with two-generated essential socle, the two simple modules in the socle are non-isomorphic, and the endomorphism ring of N is not local. Let N = g(M ). As N is a nonzero quotient of M it is couniform and, since it is a submodule of E(R/M), it has simple essential socle. We have to prove that the endomorphism ring of N is not local.
Note that rN = 0 because (1 − r)g(m) = 0 so g(m) ∈ rN and, by construction, g(m) = 0. Moreover rN = N , because otherwise M = ker(g) + rM and, since rM = M and M is couniform, we would get g = 0. Multiplication by 1 − r is a non-injective homomorphism of N that is different from zero because rN = N . As neither multiplication by r nor by 1 − r are bijective endomorphisms of N , we can conclude that the endomorphism ring of N is not local.
(ii) Let 0 = m 1 ∈ Ann M (r) and let 0 = The homomorphism g can be extended to a homomorphism g : . It is clear that N is a couniform submodule with 2-generated essential socle and that the two simple modules in the socle are non-isomorphic. Both, multiplication by r and by 1 − r, induce a nonzero homomorphism of N that is not injective. Hence the endomorphism ring of N is not local.
Proposition 3.3. Let R be a commutative ring and let M R be a couniform module with a non local endomorphism ring. Let Then, In [6, Proposition 9.23] it is proved that over a commutative noetherian ring all biuniform modules have local endomorphism ring. Here we see that this is also true for couniform modules over commutative noetherian rings. Proof. Let M be a couniform module over a commutative noetherian ring R. If the endomorphism ring of M is not local then, by Proposition 3.3, we would construct a couniform module N with essential socle over the noetherian ring R[x] with non-local endomorphism ring. This is impossible because in this case N would be artinian, and indecomposable artinian modules over commutative rings have local endomorphism ring (recall, for example, Corollary 2.6).

Category Equivalences
This section is based on the work by Pimenov and Yakovlev [25], that was further developed by Ringel in [28, §1]. The aim of these works was to construct artinian modules for which the Krull-Schmidt Theorem fails. The method was to construct category equivalences between already known classes of (noetherian) modules where the Krull-Schmidt theorem fails and classes of artinian modules. We give a general framework to these equivalences.
Throughout this section we fix a ring embedding R → T , and we let S = ( T T 0 R ).
Let S be the category of triples This defines an equivalence between the category of right S-modules and the category S. We freely use the identification between these categories.
Let R 1 f → R 2 be a ring morphism. Recall that f is said to be local if, for any r ∈ R 1 , f (r) is invertible if and only if r is invertible. We point out the following observation.
Considering ker(α A ) and coker(α A ) we obtain two functors F and G, respectively, from the category of right S-modules to the category of right R-modules. Now we describe a functor from the category of right R-modules to the category of right S-modules.
Consider the exact sequence where π denotes the canonical projection. Let M be a right R-module. Applying the functor M ⊗ R − we get the exact sequence This allows us to define a functor H from right R-modules to right S-modules by setting The 5-Lemma ensures that for any right R-module M the ring homomorphism is local.
The functor H has further properties.
(2) If T is a right Ore localization of R at a set of non-zero divisors, then H is an equivalence between C and H(C) whose inverse is F .
Proof. Statement (1) is clear because, by hypothesis, if M ∈ C then it fits into the exact sequence To see (2), we need to show that for any pair of modules M and N in C, H(Hom R (M, N )) = Hom S (H(M ), H(N )). By (1), the inclusion H(Hom R (M, N )) ⊆ Hom S (H(M ), H(N )) is always true, the hypothesis is needed to prove the reverse inclusion. Let By the universal property of the right Ore localization, f uniquely determines g 1 , so that g 1 = f ⊗ T . The universal property of the cokernel determines g 2 in a unique way.
Example 4.4. Let R be a right Ore domain with ring of quotients Q. We consider the above situation for T = Q. Fix M to be a nonzero submodule of Q. Then Notice that any element of the form (q, x) ∈ (Q, M ⊗ R Q/R; α) with q = 0 is a generator of the whole module. This implies that H(M ) is a couniform cyclic right module (i.e. it is a local right module) over S = Q Q 0 R . By Proposition 4.3, the category of torsion-free rank one modules over R is equivalent to a subcategory of local modules over the ring S. Using this it is easy to construct local modules with a pathological direct sum behavior, this makes a big difference with the situation in the commutative case. For satisfying that

Torsion free modules over noetherian rings
There is plenty of interesting literature on direct sum decompositions of torsionfree abelian groups of finite rank. Some classes of these groups have a semilocal endomorphism ring. Warfield in [30,Theorem 5.2] showed that, in general, torsionfree modules of finite rank over commutative semilocal principal ideal domains have a semilocal endomorphism ring. Our ideas allow us to extend these results to one dimensional Cohen-Macaulay commutative noetherian rings and to the noncommutative setting.
We recall that a commutative noetherian ring is one dimensional Cohen-Macaulay provided R has Krull dimension 1 and each maximal ideal contains a nonzero divisor.
In the following proposition we collect the properties of one dimensional Cohen-Macaulay rings we need. As it is seen in the proof, the statement is just a direct consequence of Matlis results on the subject.
Proposition 5.1. Let R be a semilocal commutative one dimensional Cohen-Macaulay ring. Then R has an artinian classical ring of quotients Q and K = Q/R is artinian as an R-module.
Proof. Let Q be the localization of R at the set Σ of nonzero divisor of R. The bijective correspondence between the prime ideals of Q and the prime ideals of R with no intersection with Σ implies that Q is 0-dimensional, so it is artinian.
Let In the next proposition we prove an analogous result in a non-commutative setting.
By a noetherian hereditary ring we mean a two-sided noetherian ring, hereditary on both sides.
Proposition 5.2. Let R be a semilocal hereditary noetherian ring. Then R has a (two-sided) artinian classical ring of quotients Q and K = Q/R is (serial) artinian as a right and as a left R-module.
Proof. By [23,Theorem 5.4.6], R is a finite product of artinian hereditary rings and non-artinian semilocal hereditary noetherian prime rings. Thus, to prove our claim, we may assume that R is a non-artinian HNP ring. In this situation, Goldie's Theorem implies that R has a simple artinian ring of quotients Q. Moreover, Q is the injective hull of R, both as a right R-module and as a left R-module [15,Proposition 5.13]. Being R a ring of global dimension 1, we deduce that K = Q/R in an injective R-module on both sides. We want to prove that K is artinian as a right R-module.
As R is semilocal and not artinian J(R) = 0. Since R is prime, J(R) is essential as a right (and as a left) ideal so that it contains a regular element. Because Q R is injective, we can describe the R-dual of J(R) as a right R-module as Since J(R) * is the R-dual of a finitely generated projective right R-module, it is finitely generated (and projective) as a left R-module. As J(R) = 0, all simple right and left modules are unfaithful, hence R is bounded by [12,Theorem 4.10]. Thus we can apply [21, Lemma 2.1] to deduce that there exists a regular element b ∈ R such that bJ(R) * ⊆ R. This implies that bJ(R) * and also its isomorphic copy J(R) * are finitely generated right R-modules. Let x ∈ Q. As Q is a right localization to the set of regular elements, I = {r ∈ R | xr ∈ R} contains a regular element and, hence, it is an essential right ideal of R. By [23,Proposition 5.4.5], the cyclic right submodule of Q/R, (xR + R)/R ∼ = R/I has finite length. This implies that K R has essential right socle, Soc(K R ) say. As R is semilocal By our previous arguments, K R is an injective module that has finitely generated essential right socle. Hence where, for i = 1, . . . , n, V i denotes a simple right R-module with injective hull E(V i ).
Finally, as all simple modules are unfaithful, we deduce from [14,Theorem 19] that E(V i ) is an artinian uniserial right R-module for i = 1, . . . , n. Hence K R is a serial artinian module, as we wanted to prove.
The statement on the structure of K as a left R-module follows by symmetry.
Let R be a semilocal ring that is either a one dimensional Cohen-Macaulay commutative noetherian ring or a hereditary noetherian ring. Let Q denote the classical ring of quotients of R. A right R-module M is of finite rank if M ⊗ R Q is finitely generated as right Q-module, and it is torsion free if no non-zero element of M is annihilated by a regular element of R (equivalently, if Tor R 1 (M, K) = 0). Propositions 5.1 and 5.2 give us a nice setting where to apply Proposition 4.3.
Corollary 5.3. Let R be a semilocal ring that is either a one dimensional Cohen-Macaulay commutative noetherian ring or a hereditary noetherian prime ring. Let Q denote the classical ring of quotients of Q, and let C be the category of torsion-free right R-modules of finite rank. Then any element in C has a semilocal endomorphism ring and, in fact, C is equivalent to a category of finitely generated artinian right modules over the ring S = Q Q 0 R .
Proof. Proposition 4.3, 5.1 and 5.2 allow us to conclude that the category C is equivalent to the category of modules H(C) over the ring S = Q Q 0 R . Set K = Q/R, and let M be an object of C. For any m ∈ M , the right R-module mR⊗ R K is isomorphic to a quotient of K so it is artinian (either by Proposition 5.1 or by Proposition 5.2). As M ⊗ R Q is finitely generated, there exist m 1 , . . . , is also artinian, we deduce that H(M ) is an artinian right S-module. The previous argument also shows that the right S-module H(M ) is generated by the elements As the endomorphism ring as S-module of H(M ) is isomorphic to End R (M ) and artinian modules have a semilocal endomorphism ring [2], we deduce that End R (M ) is semilocal.
6. Modules with a prescribed endomorphism ring and monoids of modules.
Let R be a ring, and let M be a right R-module. We denote by add (M ) the full subcategory of right R-modules that are isomorphic to a direct summand of a finite sum of copies of M . By Add (M ) we denote the full subcategory of right R-modules that are isomorphic to a direct summand of an arbitrary direct sum of copies of M . We recall that, by a result of Kaplansky, if M R is countably generated then any module in Add (M ) is a direct sum of countably generated modules (cf.  Assume, in addition, that M R is finitely generated. Then the functor Hom R (M, −) induces a category equivalence between Add(M R ) and the category of projective right modules over End R (M ). In particular, there is an isomorphism of monoids Remark 6.2. We follow the notation of Proposition 6.1. Set S = End R (M ). When Hom R (M, −) defines an equivalence its inverse is the functor − ⊗ S M .
Notice that, with no restriction over M and because the tensor product commutes with arbitrary direct sums, − ⊗ S M defines a functor from the category of projective right S-modules to Add (M R ). The hypothesis on M is needed to ensure the equivalence between the two categories. The precise assumption that is needed is that Hom R (M, −) commutes with arbitrary direct sums of copies of M .  As observed in [18], using Lemma 6.3 any ring that can be embedded in a local ring can be realized as endomorphism ring of a local module. For example, any domain that can be embedded in a field can be realized as endomorphism ring of a local module. Again, as with Example 4.4, this shows the big difference between the commutative and the noncommutative case.
We recall the following result from [26] which is crucial in the rest of our discussion. 6.1. The dimension monoids for semilocal rings. In this subsection, and unless otherwise is stated, R denotes a semilocal ring such that R/J(R) ∼ = M n1 (D 1 ) × · · · × M n k (D k ) for suitable division rings D 1 , . . . , D k . We fix an onto ring homomorphism ϕ : R → M n1 (D 1 ) × · · · × M n k (D k ) such that Ker ϕ = J(R).
Let V 1 , . . . , V k denote a fixed ordered set of representatives of the isomorphism classes of simple right R-modules such that End R (V i ) ∼ = D i for i = 1, . . . , k.
If P R is a countably generated projective right R-module then P/P J(R) ∼ = V (I1) 1 ⊕ · · · ⊕ V (I k ) k and the cardinality of the sets I 1 , . . . , I k determines the isomorphism class of P/P J(R). By Theorem 6.5, projective modules are determined, up to isomorphism, by its quotient modulo the Jacobson radical. So that, to describe V * (R) we only need to record the cardinalities of the sets I i for i = 1, . . . , k. Now we explain how we do that in a precise way.
Let N = {1, 2, . . . } and N 0 = N ∪ {0}. Consider also the monoid N * 0 = N 0 ∪ {∞} with the addition determined by the addition on N 0 extended by the rule n + ∞ = ∞ + n = ∞ for any n ∈ N * 0 . If P is a countably generated projective right R-module such that P/P J( Observe that dim ϕ ( R ) = (n 1 , . . . , n k ) ∈ N k . By restriction, there is also a monoid monomorphism dim ϕ : V (R) → N k 0 and its image is in a particular class of submonoids of N k 0 that we introduce in the next definition.
Definition 6.6. A submonoid A of N k 0 is said to be full affine if whenever a, b ∈ A are such that a = b + c for some c ∈ N k 0 then c ∈ A.
The class of full affine submonoids of N k 0 containing an element (n 1 , . . . , n k ) ∈ N k is the precise class of monoids that can be realized as dim ϕ (V (R)) for a semilocal ring R such that dim ϕ ( R ) = (n 1 , . . . , n k ), cf. [7].
An interesting problem is to determine which submonoids of (N * 0 ) k can be realized as dimension monoids, that is, as dim ϕ (V * (R)) for a suitable semilocal ring R. Right now it seems we are still far to be able to give an answer to this question. After [16] the answer is known in the case of noetherian rings, in the next definition we introduce the class of monoids that appears in the noetherian case. Definition 6.7. Let k ≥ 1. A submonoid B of (N * 0 ) k is said to be a monoid defined by a system of equations if it is the set of solutions in (N * 0 ) k of a system of the form where D ∈ M n×k (N 0 ), E 1 , E 2 ∈ M ×k (N 0 ), m 1 , . . . , m n ∈ N, m i ≥ 2 for any i ∈ {1, . . . , n} and , n ≥ 0.
Remarks 6.8. 1) It is important to notice that N * 0 is no longer a cancellative monoid. So that, for example, the set of solutions in (N * 0 ) 2 of the equation x = y is not the same as the set of solutions of 2x = y + x.
2) Let A be a submonoid of N k 0 . It was observed by Hochster that A is full affine if and only if it is the set of solutions in N k 0 of a system of the form (1), cf. [16, §6]. In this case, the monoid B = A + ∞ · A is a submonoid of (N * 0 ) k defined by a system of equations, cf. [16,Corollary 7.9].
For further quoting we recall the main result in [16] which characterized the monoids M that can be realized as V * (R) for a semilocal noetherian ring R. Theorem 6.9. Let k ∈ N. Let B be a submonoid of (N * 0 ) k containing (n 1 , . . . , n k ) ∈ N k . Then the following statements are equivalent: (1) B is a monoid defined by a system of equations.
(2) There exist a noetherian semilocal ring R, a semisimple ring S = M n1 (D 1 )× · · · × M n k (D k ), where D 1 , . . . , D k are division rings, and an onto ring mor- In the above statement, if F denotes a field, R can be constructed to be an F -algebra such that D 1 = · · · = D k = E is a field extension of F .
Let R be a semilocal ring such that R/J(R) ∼ = M n1 (D 1 ) × · · · × M n k (D k ) for suitable division rings D 1 , . . . , D k , and let ϕ : R → M n1 (D 1 ) × · · · × M n k (D k ) be an onto ring homomorphism such that Ker ϕ = J(R). It is not true, in general, that dim ϕ (V * (R)) is a monoid defined by a system of equations. The first problem that appears is that in the nonnoetherian setting there may be projective modules that are finitely generated modulo the Jacobson radical but that they are not finitely generated. The first example of this kind was constructed by Gerasimov and Sakhaev in [13]. A detailed study of this phenomena was done in [17].

Application to artinian modules.
It is not difficult to show that a finitely generated module over a commutative noetherian local ring S has a semilocal endomorphism ring that is a finitely generated S-algebra. Wiegand in [31] showed that if M is such a module then V (M ) can be any full affine submonoid of N k 0 having an element (n 1 , . . . , n k ) ∈ N k . This gives a nice an alternative proof of the fact that full affine monoids are the precise class of monoids that can be realized as V (R) for a semilocal ring R. It also shows that R can be taken to be a finitely generated algebra over a commutative noetherian ring. Then, using Proposition 6.1 combined with Lemma 6.3 he also proved that if N is a (cyclic) artinian module then V (N ) can be any full affine submonoid of N k 0 having an element (n 1 , . . . , n k ) ∈ N k . An alternative proof of this fact was also obtained by Yakovlev in [32].
Wiegand has two constructions of finitely generated modules, one for one dimensional rings and another for two dimensional ones. The one dimensional case fits very well in the context of Proposition 4.3 to give an alternative approach of a realization result for artinian modules. Proposition 6.10. Let A be a submonoid of N k 0 consisting on the set of solutions of a system of diophantine linear equations, and containing an element (n 1 , . . . , n k ) ∈ N k . Then there exists a one dimensional commutative local noetherian domain R with field of fractions Q such that the ring S = Q Q 0 R has an artinian module N such that V (N ) ∼ = A and this isomorphism takes N to (n 1 , . . . , n k ).
Proof. In [31] (or see also [22]), there are constructed a one dimensional local noetherian domain R and a finitely generated torsion free module M such that V (M ) ∼ = A and the isomorphism takes M to (n 1 , . . . , n k ). By Proposition 4.3, N = H(M ) has the same endomorphism as M ; so that, by Proposition 6.1, V (N ) ∼ = A and the isomorphism has the required property. Since R is one-dimensional Cohen Macaulay, N S is artinian, cf. Corollary 5.3.
Puninski in [27] was the first to observe that Add (N ), for N an artinian module, can have modules that are not direct sum of artinian ones. Again, Puninski's result is an application of Proposition 6.1 combined with Lemma 6.3. We give a more systematic approach to this phenomena by proving the following theorem, Theorem 6.11. Let k ∈ N. Let A be a submonoid of (N * 0 ) k containing (n 1 , . . . , n k ) ∈ N k . If A is a monoid defined by a system of equations then there exist a ring T and an artinian cyclic right T -module M such that V * (M ) ∼ = A.
The rest of the paper is devoted to proving Theorem 6.11 . Our strategy will be to show that we can apply Lemma 6.3 to the rings constructed to show that (1) ⇒ (2) in the proof Theorem 6.9. To do that we will need to do quite an amount of work.
We do not know whether the converse of Theorem 6.11 should be true. Endomorphism rings of artinian modules are semilocal rings satisfying the ACC on left annihilators [11]. This implies that if R is the endomorphism ring of an artinian module then any projective right R-module that is finitely generated modulo its Jacobson radical is finitely generated. That is, the situation studied in [17] cannot occur, but still we have no idea whether the monoids that could appear as V * (M ) for a cyclic artinian module M should be defined by a system of equations.

Particular classes of ring pull-backs
We examine three constructions of rings appearing in the proof of Theorem 6.9. In order to prove Theorem 6.11 we need to show that they fulfill the hypothesis of Lemma 6.3. The first one is to construct semilocal rings such that their monoid of countably generated projective modules is isomorphic to the set of solutions of a single congruence. The second one will be to construct semilocal rings such that their monoid of countably generated projective modules is isomorphic to the set of solutions of a single linear equation. The third one will show how to glue together several congruences and several equations.
All these constructions are particular classes of ring pullbacks and they come from [16,Section 5]. We first note the following easy fact.
Lemma 7.1. If R is a pullback of two rings R 1 and R 2 that can be embedded in artinian rings then also R embeds in an artinian ring.
Proof. If for i = 1, 2, R i embeds in the artinian ring S i then R embeds in S 1 × S 2 which is an artinian ring.
Next construction does some preliminary work needed to construct the bimodule required in the statement of Lemma 6.3.
Construction 7.2. Let F be a field. Let R = M n1 (F ) × · · · × M n k (F ) and S = M m (F ). Assume that (a 1 , . . . , a k ) ∈ N k 0 is such that a 1 n 1 + · · · + a k n k = m. Let α : R → S be the ring homomorphism given by So that α induces an R-R-bimodule structure over S. Let X ∈ M m (F ). Fix ∈ 1, . . . , k, for i ≤ a and 1 ≤ i, let X i be the submatrix of X determined by the entries that are in the intersection of the n rows ranging from a 1 n 1 + · · · + a −1 n −1 + (i − 1)n + 1 and a 1 n 1 + · · · + a −1 n −1 + (i)n and the n columns ranging from a 1 n 1 + · · · + a −1 n −1 + (i − 1)n + 1 and a 1 n 1 + · · · + a −1 n −1 + (i)n .
Consider the map α * : S → R given by α * (X) = ( , and consider the maps α * A : S → R given by α * A (X) = α * (AX) and β A : S → R given by (X)β A = α * (XA). Lemma 7.3. With the notation above, α * is a morphism of R-R-bimodules, α * A is a morphism of right R-modules and β A is a morphism of left R-modules.
In view of Lemma 7.3, there are maps Φ : S → Hom R (S R , R R ) and Φ : S → Hom R ( R S, R R) given by Φ(A) = α * A and Φ (A) = β A for any A ∈ M m (F ).
Lemma 7.4. Φ is an isomorphism of right S-modules and Φ is an isomorphism of left S-modules. Let γ : Hom R (S R , R) → R be given by γ(f ) = f (1), and let γ : Hom R ( R S, R) → R be given by γ(g) = g (1).
. The associativity of the product of matrices yields that Φ(AB) = α * AB = (α * A )B = Φ(A)B. Therefore Φ is a morphism of right S-modules.
Since dim F (S) = dim F (Hom R (S R , R)), to conclude that Φ is an isomorphism it is enough to show that it is injective. Let 0 = A ∈ S, let 1 ≤ i, j ≤ m be such that the i-j-entry of A is different from zero. Let E ji ∈ S be such that all its entries are zero except from the j-i-entry which is one, then α * A (E ji ) = α * (AE ji ) = 0 so that Φ(A) = 0. This shows that Φ(A) = 0 if and only if A = 0 and, hence, Φ is injective. An easy computation shows that γ • Φ = α * .
The statements for Φ are proved in a similar way. Assume (n 1 , . . . , n k ) ∈ N k is such that a 1 n 1 + · · · + a k n k = m ∈ N. Let F be a field. Assume that there exists a semilocal principal ideal domain R 1 such that R 1 /J(R 1 ) ∼ = M m (F ) and that J(R 1 ) is generated by a central element of R 1 . Fix an onto ring homomorphism j : M (R 1 ) → M m (F ) with kernel J(M (R 1 )) = M (J(R 1 )).
Set R 2 = M n1 (F ) × · · · × M n k (F ), and consider the morphism α : Let R be the ring defined by the pullback diagram Then R is noetherian semilocal, it embeds into an artinian ring, Ker ϕ = J(R) and ϕ is onto. Hence, ϕ induces an isomorphism R/J(R) ∼ = M n1 (F ) × · · · × M n k (F ). Moreover, dim ϕ V * (R) is exactly the set of solutions in (N * 0 ) k of the congruence a 1 t 1 + · · · + a k t k ∈ mN * 0 .
Lemma 7.6. In the situation and notation of Construction 7.5, assume that X is a right module over M (R 1 ). Then is of finite length, then X R is also a module of finite length.
(ii) If X is an artinian M (R 1 )-module, then it is also artinian as R-module.
Proof. (i). Let V be the simple right module over M (R 1 ). Since V M (J(R 1 )) = 0, V is a simple M m (F )-module, and hence a module of finite length over α(R 2 ). Therefore, it is also a module of finite length over R 2 .
Since J(R) = M (J(R 1 )) × {0}, V J(R) = 0. So that the structure of V as R-module is the same as the structure of V as R 2 -module. Hence, V is a module of finite length over R.
The statement for a general module over M (R 1 ) of finite length follows easily by induction on the composition length.
(ii). Let be a descending chain of right R-submodules of X. Since i(J(R)) = J(M (R 1 )) is a descending chain of right M (R 1 )-submodules of X; therefore there exists n 0 such that X n0 J(R) = X n0+k J(R) for any k ≥ 0. Consider now the descending chain Since Y is an artinian module over M (R 1 )/M (J(R 1 )) it is of finite length. By (i), Y is an R-module of finite length. Therefore there exists n 1 ≥ n 0 such that Then it follows that X n1 = X n1+k for any k ≥ 0. This proves that X R is artinian.
Following the notation of Construction 7.5, let Q denote the field of fractions of R 1 . Then Q/J(R 1 ) ∼ = Q/R 1 is an R 1 -R 1 -bimodule which is an injective cogenerator and artinian on both sides (apply, for example, Proposition 5.2). The hypothesis on the PID R 1 ensure that the right and left socle of Q/J(R 1 ) coincide with the R 1 -R 1subbimodule R 1 /J(R 1 ). Therefore M (Q/J(R 1 )) is an M (R 1 )-M (R 1 )-bimodule which is an artinian injective cogenerator on both sides, and its right and left (essential) socle coincide with M (R 1 /J(R 1 )). Let : M (R 1 /J(R 1 )) → M m (F ) be the isomorphism induced by the homomorphism j. Consider the following push-out of abelian groups where α * denotes the map from Construction 7.2 associated to the map α in the pull-back diagram (2) defining R.
Proposition 7.7. With the notation above, the following statements hold: (iv) N is an injective R-cogenerator on both sides.
(v) N is an artinian R-module on both sides.
As remarked before, the M (R 1 )-bimodule M (Q/J(R 1 )) is injective on both sides and serial artinian on both sides by Proposition 5.2. By the way the ring R 1 is chosen the right and left socle of M (Q/J(R 1 )) is M (R 1 /J(R 1 )) which coincides with −1 (M m (F )).
(i). To prove that N is an R-R bimodule we must see that L is invariant, on both sides, by the action of R. Let A ∈ R 1 and B ∈ R 2 be such that (A, B) (Xj(A)), −α * (X)B). Now α * (Xj(A)) = α * (Xα(B)) = α * (X)B. This proves that L is a right R-module. Similarly, it also follows that L is a left R-module.
(iii). Clearly, ε(R 2 ) is an R-R-submodule of N which is semisimple on both sides. Therefore it is contained in the right socle of N and in the left socle of N . To prove that it coincides with both socles we shall see that ε(R 2 ) is essential in N as a right and as a left R-module.
Indeed, A = (a ij + J(R 1 )) with a ij ∈ Q for any i, j ∈ {1, . . . , }. Choose i 0 , j 0 such that a i0j0 ∈ Q \ R 1 . Then, since Q/J(R 1 ) has essential socle R 1 /J(R 1 ), there exists x ∈ J(R 1 ) such that a i0j0 x ∈ R 1 \ J(R 1 ). Let D be the matrix of M (J(R 1 )) with all its entries zero except for the entry j 0 -i 0 which is x. Then in the matrix AD ∈ M (Q/J(R 1 )) only the i 0 column is non-zero and the i 0 -i 0 entry of AD is 0 = a i0j0 x. If all the entries of AD are in R 1 + J(R 1 ) we set C = D otherwise we repeat the above process with an entry of AD which is not in R 1 + J(R 1 ). At the end we get a matrix 0 = AD 1 · · · D r ∈ M (R/J(R 1 )) with only one non-zero column and a nonzero entry in the diagonal. Moreover D 1 · · · D r ∈ M (J(R 1 )), so that j(D 1 · · · D r ) = 0 and, hence, (D 1 · · · D r , 0) ∈ R.
There exists X ∈ M m (F ) such that AD 1 · · · D r = −1 (X). Notice that such X will have only one non-zero column and a nonzero entry in the diagonal. Therefore α * (X) = 0. We finish the proof of the claim setting C = D 1 · · · D r .
A symmetric argument shows that ε(R 2 ) is also an essential left submodule of N , hence it also coincides with the left socle of N .
(iv). The injectivity of N , on both sides, follows from [9, Theorem 1]. We briefly explain how we apply this result.
Since R 2 is semisimple it is injective as R 2 -module on both sides, also M (Q/J(R 1 )) is an M (R 1 )-M (R 1 )-bimodule which is injective on both sides and its socle is isomorphic via to M m (F ). By Lemma 7.4, there are isomorphisms M m (F ) → Hom R2 (M m (F ), R 2 ) of right and of left M m (F )-modules and both isomorphisms composed with the evaluation at the identity give the map α * . Therefore the pushout in (6) is of the type in [9, p. 427] so it gives an R-bimodule that is injective on both sides.
By (iii), N is the injective hull of R/J(R) on both sides. Since R is semilocal, it is an injective cogenerator on both sides.
(v). In view of the identity (7), N/ε(R 2 ) ∼ = M (Q)/M (R 1 ). So that there is an exact sequence is semisimple artinian as a left and as a right R-module it suffices to show that M (Q)/M (R 1 ) is artinian both as a right and as a left Rmodule. Recall that, by Proposition 5.2, X = M (Q)/M (R 1 ) is artinian, on both sides, as R 1 -module, and, by Lemma 7.6, it is also artinian, on both sides, as Rmodule. This allows us to conclude that N is artinian, on both sides, as R-module. N 0 . Let (n 1 , . . . , n k ) ∈ N k be such that a 1 n 1 +· · ·+a k n k = b 1 n 1 +· · ·+b k n k = m ∈ N. Let F be a field. Let Let j 1 : M m (R 1 ) → M m (F )×M m (F ) be an onto ring homomorphism with kernel J(M m (R 1 )). Set R 2 = M n1 (F )×· · ·×M n k (F ). Consider the morphism j 2 : R 2 −→ M m (F ) × M m (F ) defined by j 2 (r 1 , . . . , r k ) = (α 1 (r 1 , . . . , r k ), α 2 (r 1 , . . . , r k )) = Let R be the ring defined by the pullback diagram Then R is a noetherian semilocal F -algebra, it embeds into an artinian ring and ϕ is an onto ring homomorphism with kernel J(R). Moreover, dim ϕ V * (R) is the set of solutions in (N * 0 ) k of the equation a 1 t 1 + · · · + a k t k = b 1 t 1 + · · · + b k t k .
Following the notation of Construction 7.8. Let Q denote the field of fractions of R 1 . Then Q/J(R 1 ) ∼ = Q/R 1 is an R 1 -R 1 -bimodule which is an artinian injective cogenerator on both sides by Proposition 5.2. The right and left socle of Q/J(R 1 ) coincide with the R 1 -R 1 -subbimodule R 1 /J(R 1 ). Therefore M m (Q/J(R 1 )) is an M m (R 1 )-M m (R 1 )-bimodule which is injective and artinian on both sides, and its right socle and left socle coincide with M m (R 1 /J(R 1 )). Let 1 : M m (R 1 /J(R 1 )) → M m (F ) × M m (F ) be the isomorphism induced by the homomorphism j 1 . Consider the following push-out of abelian groups where δ(X, Y ) = α * 1 (X) + α * 2 (Y ) for any (X, Y ) ∈ M m (F ) × M m (F ). Here, for i = 1, 2, α * i denotes the map associated to α i in Construction 7.2. Following the same ideas as in the proof of Proposition 7.7 we have the following properties for N . Proposition 7.9. With the notation above, the following statements hold: 7.3. Adding equations and congruences. Finally, we need the following construction.
Construction 7.10. Let (n 1 , . . . , n k ) ∈ N k . Let F be a field. Assume that there exist semilocal notherian rings R 1 and R 2 with fixed onto morphisms ϕ i : R i → M n1 (F ) × · · · × M n k (F ) such that Ker ϕ i = J(R i ) for i = 1, 2. Assume that, for i = 1, 2, dim ϕi (V * (R i )) is the set of solutions in (N * 0 ) k of a system of the form . . , m i ni ∈ N, m i j ≥ 2 for any j ∈ {1, . . . , n i } and i , n i ≥ 0. Consider the pullback Then R is a semilocal noetherian ring. The morphism ϕ = ϕ 1 i 1 = ϕ 2 i 2 is onto, Ker ϕ = J(R), and dim ϕ (V * (R)) is the set of solutions in (N * 0 ) k of the system S 1 S 2 .
Proposition 7.11. We follow the notation of Construction 7.10. For i = 1, 2, let N i be a right R i -module with a fixed embedding of right R i -modules ε i : M n1 (F ) × · · · × M n k (F ) → N i . Consider the push-out of abelian groups Then: (i) N is a right R-module and ε = π 1 ε 1 = π 2 ε 2 is injective.
(iii). Since, for j = 1, 2, the morphism i j in Construction 7.10 is onto, if M is an R i -module then its lattice of R-submodules is exactly the same as its lattice of R i -submodules. Then the claim follows easily from diagram (10) because N fits into the exact sequence of R-modules 0 → N 1 → N → N 2 /Im ε 2 → 0.
(iv). As with the other constructions, it follows from the results in [9] that N is an injective right R-module. An alternative proof can be done following word by word the argument of [5, Theorem 3.1(3)] to conclude the injectivity of N . Now we are ready to prove Theorem 6.11. Proof Theorem 6.11. Let A be a submonoid of (N * 0 ) k that is the set of solutions in (N * 0 ) k of the system where D ∈ M n×k (N 0 ), E 1 , E 2 ∈ M ×k (N 0 ), m 1 , . . . , m n ∈ N, m i ≥ 2 for any i ∈ {1, . . . , n} and , n ≥ 0. We may assume that either of n is > 0 By [16, Example 3.3(i)], for any t ∈ N, we can construct a semilocal PID R such that R/J(R) ∼ = M t (Q) and J(R) is generated by a central element. Therefore, using Construction 7.5, by i = 1, . . . n we can construct noetherian semilocal rings R i with an onto ring homomorphism ϕ i : R i → M n1 (Q) × · · · × M n k (Q) such that Ker ϕ i = J(R i ), and satisfying that dim ϕi (V * (R i )) is the set of solutions of the i-th congruence in the system defining A.
Using Construction 7.8, for j = 1, . . . , we can construct noetherian semilocal rings R n+j with an onto ring homomorphism ϕ n+j : R n+j → M n1 (Q)×· · ·×M n k (Q) such that Ker ϕ n+j = J(R n+j ) satisfying that dim ϕn+j (V * (R n+j )) is the set of solutions of the j-th linear equation in the system defining A.
By construction, all these rings can be embedded in suitable artinian rings. In view of Proposition 7.7 and Proposition 7.9, for i = 1, . . . , n + , there exists an R i -R i -bimodule N i and an embedding of R i bimodules ε i : M n1 (Q)×· · ·×M n k (Q) → N i such that (i) N i is artinian on both sides as R i -module. (ii) N i is an injective R i -module on both sides and its socle, on both sides, is Im ε i . (iii) Ri (N i ) Ri is a cogenerator on both sides.
A successive application of Construction 7.10 with the homomorphisms ϕ 1 , . . . , ϕ n+ yields a semilocal noetherian ring R that can be embedded in an artinian ring, an onto homomorphism ϕ : R → M n1 (Q) × · · · × M n k (Q) such that Ker ϕ = J(R) and such that dim ϕ (V * (R)) = A. By Proposition 7.11 and its left handed version, taking the corresponding successive pushouts of ε 1 , . . . , ε n+ we obtain an R-R-bimodule N and an embedding ε : M n1 (Q) × · · · × M n k (Q) → N such that (i') N is artinian on both sides as R-module. (ii') N is an injective R-module on both sides and its socle, on both sides, is Im ε. (iii') R N R is a cogenerator on both sides. By Lemma 6.3, R can be realized as the endomorphism ring of an artinian cyclic module M such that V * (M ) ∼ = A. This concludes the proof of the theorem. Now we discuss some examples to illustrate Theorem 6.11. Example 7.12. In view of Theorem 6.11, there exists a cyclic artinian module M 1 such that V * (M 1 ) ∼ = A 1 = {(x, y) ∈ (N * 0 ) 2 | x = y}. Since A 1 = (1, 1)N * 0 it follows that Add(M 1 ) contains, up to isomorphism, a single indecomposable module and that any module in Add(M 1 ) is isomorphic to a direct sum of copies of this indecomposable module.