Further results on skew monoid rings of a certain free monoid (I)

Abstract Let R be a ring with an endomorphism , F be a free monoid and S be a factor of such that for some positive integer . The second author and Moussavi [Annihilator properties of skew monoid rings, Comm. Algebra, 42 (2) (2014), 842–852] started studying the skew monoid ring . In this paper, we continue the study of these rings.


Introduction
Throughout this article, all rings are associative with identity, and 1 will always stand for the identity of the monoid S and the ring R. For a ring R, we denote by U(R) and J(R) the set of invertible elements and the Jacobson radical of R, respectively. For a non-empty subset X ⊂ R, r R (X) and R (X) denote the right and left annihilators of X in R, respectively. Also Z (R) and Z r (R) denote the left and right singular ideals of R, respectively. Let n ≥ 2 be a positive integer number and S be a free monoid generated by = {u 1 , … , u t } with 0 added, and the relations u i 1 … u i n = 0 for all 1 ≤ i 1 , … , i n ≤ t. Note that maybe u i 1 … u i k is equal to zero or not, for k < n but the product of n elements of is certainly zero. Let R be a ring with an endomorphism with (1) = 1. Then is a ring with usual addition and multiplication subject to the relation R[S; ] = {r 1 s 1 + ⋯ + r k s k : r i ∈ R , s i ∈ S , k ∈ ℕ} u i r = (r)u i ,

PUBLIC INTEREST STATEMENT
This research is devoted to Non-commutative algebra and Non-commutative Representation Theory. Mohammad Habibi and Ahmad Moussavi [Annihilator properties of skew monoid rings, Comm. Algebra, 42 (2) (2014), 842-852] started introducing and studying the special skew monoid ring. In this work, the authors continue the study of these rings.
for each u i ∈ and r ∈ R. The skew monoid ring R[S; ] was introduced by Habibi and Moussavi (2014). They characterized various radicals of R[S; ]. These rings are perhaps the most interesting class of non-semiprime rings and may provide many surprising examples and counterexamples in ring theory. Motivated by results in Habibi and Moussavi (2014), Paykan (2017), and Paykan and Arjomandfar (2017), we will obtain criteria for R[S; ] to satisfy various conditions on rings.
The main purpose of this paper is to continue the study of the skew monoid rings R[S; ]. In Section 2, we give some examples of these rings in order to familiarize the reader with the concept. In this section, the reader shall realize the importance of this structure in matrix theory. In Section 3, we characterize when R[S; ] is left quasi-duo, clean, exchange, Hermite, semiregular and I-ring, respectively. Also, we calculate the right singular ideal of R[S; ] under appropriate conditions. As a corollary we show that, R is right weakly continuous if and only if R[S; ] is right weakly continuous. In particular, prove that several properties, including the semiboolean, stable finite, 2-good, and stable range one property, transfer between R and the extension R[S; ]. Also, we show that R[S; ] is a (strongly) nil-clean ring if and only if R is a (strongly) nil-clean ring. As an application, we provide (apparently) new examples of the aforementioned ring constructions.

Examples
For a positive integer n, a ring R and an endomorphism , in Chen, Yang and Zhou (2006) defined the skew triangular matrix ring T n (R, ) as the set of all upper triangular n × n matrices over R with pointwise addition and a new multiplication subject to the condition where E ij is the matrix unit for each i and j. So for any (a ij ) and where for each i ≤ j. The subring of the ring T n (R, ) consisting of triangular matrices with constant main diagonal is denoted by S(R, n, ), whereas the subring of T n (R, ) consisting of triangular matrices with constant diagonals is denoted by T(R, n, ). We can denote A = (a ij ) ∈ T(R, n, ) by (a 11 , a 12 … , a 1n ). Then T(R, n, ) is a ring with pointwise addition and with multiplication given by: We consider the following two subrings of S(R, n, ), as follow (see Habibi, Moussavi, & Mokhtari, 2012): For example: Below we show how the aforementioned classical ring constructions can be viewed as special cases of the skew monoid ring construction of a certain free monoid.
Example 2.1 Let R be a ring with an endomorphism and u be a matrix in T n (R, ) as follows: So we have for each 2 ≤ l ≤ n − 1 and u n = 0. Let S be a free monoid generated by = {u} with 0 added and the relation u n = 0. Thus where I n is the identity matrix of order n. Therefore, we have So and hence R[S; ] = T(R, n, ).

Example 2.2
Let R be a ring with an endomorphism . It is not hard to see that ] is the skew polynomial ring with multiplication subject to the condition xr = (r)x for each r ∈ R, and ⟨x n ⟩ is the ideal generated by x n . So R[x; ]∕⟨x n ⟩ ≅ T(R, n, ) and consequently R[x; ]∕⟨x n ⟩ fit into the structure introduced in introduction.

Example 2.3
Let R be a ring with an endomorphism , and let S be a free monoid generated by = {u, v} with 0 added and the relation a 0 a 1 ⋯ a n−2 a n−1 Example 2.4 Let R be a ring with an endomorphism and n ≥ 2 be a positive integer number. Assume that S is a free monoid generated by the set of all elements E i,i+1 where i = 1, … , n − 1 with 0 added and the relations E i 1 ,i 1 +1 … E i n ,i n +1 = 0 for all 1 ≤ i 1 , … , i n ≤ n − 1. Then we have and hence R[S; ] = S(R, n, ).
Example 2.5 Let R be a ring with an endomorphism and n ≥ 2 be a positive integer number. Suppose u = E 1,2 + ⋯ + E n−1,n is a matrix in T n (R, ) as mentioned as in Example 2.1. Let is a free monoid generated by with 0 added, and that S is a factor of F setting certain monomials in to 0, enough so that, (S ⧵ {1}) n = 0. Then we have and hence R[S; ] = A(R, n, ).

Example 2.6
Let R be a ring with an endomorphism and n = 2k be a positive integer number. Let = {u, E 1,k , E 1,k+1 + E 2,k+2 , … , E n−k,n }. Now assume F ∪ {0} is a free monoid generated by with 0 added, and that S is a factor of F setting certain monomials in to 0, enough so that, (S ⧵ {1}) n = 0. Then, we have and hence R[S; ] = B(R, n, ).
The above examples mentioned only for familiarizing the reader with the structure R[S; ].
Although various examples of this structure exist in the Pure algebra, but as far as in the above example was observed, only applied examples are related to matrix theory. Therefore, the results obtained in this paper can be widely used in matrix theory.

Main Results
Recall that a ring R is local if R/J(R) is a division ring, and R is semilocal if R/J(R) is a semisimple ring. A ring R is said to be matrix local if R/J(R) is a simple Artinian ring. A ring R is said to be semiperfect if R is a semilocal ring and all idempotents of the Artinian ring R/J(R) can be lifted to idempotents of the ring R. Due to Nicholson (1975), a ring R is said to be an I-ring if every right ideal of the ring R not contained in J(R) contains a nonzero idempotent and all idempotents of the ring R/J(R) can be lifted to idempotents of the ring R. Recall from Nicholson (1976) that a ring R is semiregular if R/J(R) is a (von Neumann) regular ring and idempotents can be lifted modulo J(R). According to Nicholson (2004), a ring R is called a clean (uniquely clean) ring if every element r ∈ R can be written (uniquely) in the form r = u + e where u is a unit in R and e 2 = e ∈ R.
Throughout this section, we assume that all the monoids S is a free monoid generated by natural number. We start with the following lemma, which plays a key role in the sequel.
Lemma 3.1 Let R be a ring and an endomorphism of R. Then Clearly, this implies that has a right inverse in T. Analogously as above, one can show that has a left inverse in T and thus ∈ U(T). Conversely, if is an element of U(T), then one can easily verify that a ∈ U(R) and the result follows. ✷

Lemma 3.2 [Habibi & Moussavi, 2014 Theorem 2.9(i)] Let R be a ring and an endomorphism of R. Then
Proof Assume that = ∑ s∈S r s s is an element of T = R[S; ] with r 1 ∈ J(R) and = ∑ s∈S p s s an arbitrary element of T. Thus 1 − p 1 r 1 ∈ U(R) and by Lemma 3.1, 1 − ∈ U(T) which means that ∈ J(T). Conversely, suppose that ∈ J(T). So 1 − a ∈ U(T) for each a ∈ R and hence 1 − ar 1 ∈ U(R), by Lemma 3.1. Therefore, r 1 ∈ J(R) and we are done. ✷

Lemma 3.3 Let R be a ring and an endomorphism of R, and T = R[S;
]. Then, we have the following: (1) The factor ring R/J(R) is naturally isomorphic to the factor ring T/J(T). (

2) All idempotents of T/J(T) can be lifted to T if and only if all idempotents of the factor ring R/J(R)
can be lifted to R.
Proof It is easy to show that the map : (2) The result follows from (1). ✷ Theorem 3.4 Let R be a ring and an endomorphism of R. Then: ] is a local ring if and only if R is a local ring.
(2) R[S; ] is a semilocal ring if and only if R is a semilocal ring.
(3) R[S; ] is a matrix local ring if and only if R is a matrix local ring. (4) This result is a direct consequence of (2) and Lemma 3.3.
(5) By [Lam, 1991, Theorem 23.10] the class of all matrix local semiperfect rings coincides with the class of all rings that are isomorphic to full matrix rings over local rings. Therefore, part (5) follows from (3) and (4).
(6) The result follows from [Nicholson, 1975, Proposition 1.4] and Lemma 3.3, since R is an I-ring if and only if R/J(R) is an I-ring and all idempotents of the ring R/J(R) can be lifted to idempotents of the ring R.
(7) This result is a consequence of Lemma 3.3.
(8) The result follows from [Camillo & Yu, 1994, Proposition 7] and Lemma 3.3, since R is a clean ring if and only if R/J(R) is a clean ring and all idempotents of the ring R/J(R) can be lifted to idempotents of the ring R. ✷ A ring R is called right (left) quasi-duo if every maximal right (left) ideal of R is two-sided or, equivalently, every right (left) primitive homomorphic image of R is a division ring. Examples of right quasiduo rings include, for instance, commutative rings, local rings, rings in which every non-unit has a (positive) power that is central, endomorphism rings of uniserial modules, power series rings and rings of upper triangular matrices over any of the above-mentioned rings (see Yu, 1995). But the nby-n full matrix rings over right quasi-duo rings are not right quasi-duo ( for more details see Lam & Alex, 2005;Leroy, Matczuk, & Puczylowski, 2008;Yu, 1995).
A ring R is said to be Dedekind finite if ab = 1 implies ba = 1 for any a, b ∈ R; and R is stably finite if any matrix ring M n (R) is Dedekind-finite (for more details see Montgomery, 1983). Recall that a module R M has the (full) exchange property if for every module R A and any two decompositions . A module R M has the finite exchange property if the above condition is satisfied whenever the index set I is finite. Warfield (1972) called a ring R an exchange ring if the left regular module R R has the finite exchange property and showed that this definition is left-right symmetric.

Recall that a ring R is semiboolean if and only if R/J(R) is Boolean and idempotents of R lift modulo J(R).
According to [Nicholson & Zhou, 2004, Theorem 19], R is a Boolean ring if and only if R is uniquely clean and J(R) = 0. By Lemma 3.2, for an arbitrary (Boolean) ring R, the ring R[S; ] is not Boolean. But we will show that R[S; ] is semiboolean if and only if R is semiboolean.
According to Vámos (2005), a ring R is said to be 2-good if every element is the sum of two units. The ring of all n-by-n matrices over an elementary divisor ring is 2-good (where the ring R is elementary divisor if, for every positive integer n, every element of all n-by-n matrices with entries from R, is equivalent to a diagonal matrix). A (right) self-injective von Neumann regular ring is 2-good provided it has no 2-torsion. In Wang and Ren (2013), showed that the 2-good property is preserved in extensions such as skew power series rings, full matrix rings, formal triangular matrix rings, upper triangular matrix rings, and trivial extension rings. (for more details see Vámos, 2005; Wang & Ren, 2013).

Theorem 3.5 Let R be a ring and an endomorphism of R. Then:
(

1) R[S; ] is left quasi-duo if and only if R is left quasi-duo.
(2) R[S; ] is stably finite if and only if R is stably finite. Proof (1) The result follows from Lemma 3.3, since a ring R is left quasi-duo if and only if so is R / J(R).
(2) Let T = R[S; ] be stably finite. Clearly, the subring R is also stably finite. Conversely, suppose that R is stably finite. Consider the ideal < u 1 , … , u t > in T. By Lemma 3.2, < u 1 , … , u t >⊆ J(T). We have T∕ < u 1 , … , u t >≅ R, so by [Montgomery, 1983, Lemma 2], the fact that R is stably finite implies that T is stably finite.
(3) The result follows from [Nicholson, 1977, Corollary 2.4 (2), we can show that the ring R is a homomorphic image of R[S; ]. By [Wang & Ren, 2013, Proposition 2.15], every homomorphic image of a 2-good ring is again 2-good, and therefore, the result follows. ]. Let I 0 be the set of all leading coefficients of elements of I. Then I 0 is a right ideal of R. On the other hand, we also have r 1 I 0 = 0. We show that I 0 is an essential right ideal of R. Let a be an arbitrary nonzero element of R. There exists ∈ R[S � ; ] such that 0 ≠ a ∈ I. Hence, 0 ≠ ab ∈ I 0 for some b ∈ R. Thus I 0 is an essential right ideal of R. So r 1 ∈ Z r (R). Conversely, let r 1 ∈ Z r (R). There exists an essential right ideal J of R such that r 1 J = 0. We show that I = Ju n−1 is an essential right ideal of R[S � ; ]. Assume that = b 1 + b 2 u + b 3 u 2 + ⋯ + b n u n−1 is an arbitrary nonzero element of R[S � ; ]. Also, let j be the minimum index such that b j ≠ 0. There exists y ∈ R such that 0 ≠ b j y ∈ J. Since is an epimorphism of R, there is x such that j−1 (x) = y. Set : = xu n−j−1 ∈ R[S  Corollary 3.8 Let R be a ring and an epimorphism of R. If R is right non-singular, then Z r (R[S � ; ]) is nilpotent.
A ring R is called right weakly continuous if R is semiregular and Z r (R) = J(R). By [Nicholson & Yousif, 2003, Theorem 7.40] right weakly continuous is a Morita invariant property of rings. In the following Theorem 3.9, we prove that the right weakly continuous property of R preserves by R[S � ; ]. From Cohn (cohn), a ring R is said to be projective-free if every finitely generated projective left (equivalently right) R-module is free of unique rank. A ring homomorphism is called local if every non-unit is mapped to a non-unit.

Theorem 3.11 Let R be a projective-free ring and an endomorphism of R. Then R[S;
] is a projectivefree ring.
Proof There is a ring epimorphism :R[S; ] → R which sends ∑ s∈S a s s to a 1 . By Lemma 3.1, it follows that is local. By [Cohn, 2003, Corollary 4], any ring with a surjective local homomorphism to a projective-free ring is projective-free, and so, the result follows. ✷ As an immediate consequence of Theorem 3.11, we obtain the following.   (1,0) Since R is a Hermite ring, there exists a m × m matrix over R with first row (a (1,0) , a (2,0) , … , a (m,0) ) and det(P) ∈ U(R). Let Then, it is easy to see that the constant coefficient of det(Q) is equal to det(P) and thus in U(R). Hence, by Lemma 3.1, it follows that det(Q) ∈ U(R[S; ]). This proves that R[S; ] is a Hermite ring. ✷ Recall that a R-module P is a stably free module if there exist m, n ∈ ℕ such that P ⊕ R m = R n .
Clearly free modules are stably free. It is well known that R is a Hermite ring if and only if every stably free R-module is free (see, for example, Zhou, 1988, pp. 357-358). Thus, by Theorem 3.13, we have: Corollary 3.14 The following conditions are equivalent: (1) Every stably free R-module is free.
Recall that a module is said to be uniserial if any two of its submodules are comparable with respect to inclusion, i.e. any two of its cyclic submodules are comparable by set inclusion. A ring R is called a right (resp. left) uniserial ring, if R R (resp. R R) is a uniserial module. Right uniserial rings are also called right chain rings, or right valuation rings, since they are obvious generalizations of In the following, we state that the converse of Theorem 3.15 is not true, in general.
Example 3.16 Let K be a field with an automorphism , and let S be a free monoid generated by = {u, v} with 0 added and the relation Thus K is simple, as a right K-module, but the submodules Ku and Kv are not comparable with respect to inclusion.
Theorem 3.17 Let R be a ring and an epimorphism of R, and M a right R-module. Then the following conditions are equivalent: (1) M is a simple right R-module.
Proof (1) ⇒ (2) Suppose = p 1 + p 2 u + ⋯ + p n u n−1 is elements of M[S � ]. Let j be an smallest index with property p j ≠ 0. Hence p j R = M, since M is a simple right R-module. Thus, Now, let 1 = a 1 + a 2 u + ⋯ + a n u n−1 and 2 = b 1 + b 2 u + ⋯ + b n u n−1 be two elements of M[S � ]. Without loss of generality, one can assume that j 1 ≤ j 2 , where j 1 ( j 2 ) is an smallest index of 1 ( 2 ) with property a j 1 ≠ 0 (b j 2 ≠ 0). A ring R is called a right (resp. left) serial ring, if R R (resp. R R) is a serial module. A ring R is called a serial ring, if R is both a right and a left serial ring. It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975), such a ring is a direct sum of an Artinian ring and hereditary prime rings.