On a class of semicommutative modules

Kyungpook Mathematical Journal

We say a module a semicommutative module if for any and any , implies . This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and be a p.p.-module, then is a semicommutative module iff is an Armendariz module. For any ring R, R is semicommutative iff A(R, ) is semicommutative. Let R be a reduced ring, it is shown that for number and , is semicommutative ring but is not.

2016

Abstract. We say a module MR a semicommutative module if for any m ∈ M and any a ∈ R, ma = 0 implies mRa = 0. This paper gives various properties of reduced, Ar-mendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and MR be a p.p.-module, then MR is a semicommutative module iff MR is an Armendariz module. For any ring R, R is semicommutative iff A(R,α) is semicommu-tative. Let R be a reduced ring, it is shown that for number n ≥ 4 and k = [n/2], T kn (R) is semicommutative ring but T k−1n (R) is not. 1.

Acta Universitatis Sapientiae, Mathematica, 2018

Let R be a ring, MR a module, S a monoid, ω : S → End(R) a monoid homomorphism and R * S a skew monoid ring. Then M[S] = {m1g1 + · · · + mngn | n ≥ 1, mi ∈ M and gi ∈ S for each 1 ≤ i ≤ n} is a module over R ∗ S. A module MR is Baer (resp. quasi-Baer) if the annihilator of every subset (resp. submodule) of M is generated by an idempotent of R. In this paper we impose S-compatibility assumption on the module MR and prove: (1) MR is quasi-Baer if and only if M[s]R∗S is quasi-Baer, (2) MR is Baer (resp. p.p) if and only if M[S]R∗S is Baer (resp. p.p), where MR is S-skew Armendariz, (3) MR satisfies the ascending chain condition on annihilator of submodules if and only if so does M[S]R∗S, where MR is S-skew quasi-Armendariz.

Kyungpook mathematical journal, 2009

For an endomorphism α of R, in [1], a module MR is called α-compatible if, for any m ∈ M and a ∈ R, ma = 0 iff mα(a) = 0, which are a generalization of α-reduced modules. We study on the relationship between the quasi-Baerness and p.q.-Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [2] and some results in [9]. In particular, we show: for an α-compatible module MR (1) MR is p.q.-Baer module iff M [x; α] R[x;α] is p.q.-Baer module. (2) for an automorphism α of R, MR is p.q.-Baer module iff M [x, x −1 ; α] R[x,x −1 ;α] is p.q.-Baer module.

Journal of Algebra, 2019

Baer rings and modules Quasi-Baer rings and modules p.q.-Baer modules Extending and FI-extending modules Endomorphism rings Annihilators Semicentral Idempotents In this paper it is shown that, for a module M over a ring R with S = End R (M), the endomorphism ring of the R[x]-module M [x] is isomorphic to a subring of S[[x]]. Also the endomorphism ring of the R[[x]]-module M [[x]] is isomorphic to S[[x]]. As a consequence, we show that for a module M R and an arbitrary nonempty set of not necessarily commuting indeterminates X, M R is quasi-Baer if and only if M [X] R[X] is quasi-Baer if and only if M [[X]] R[[X]] is quasi-Baer if and only if M [x] R[x] is quasi-Baer if and only if M [[x]] R[[x]] is quasi-Baer. Moreover, a module M R with IFP, is Baer if and only if M [x] R[x] is Baer if and only if M [[x]] R[[x]] is Baer. It is also shown that, when M R is a finitely generated module, and every semicentral idempotent in S is central, then M [[X]] R[[X]] is endo-p.q.-Baer if and only if M [[x]] R[[x]] is endo-p.q.-Baer if and only if M R is endo-p.q.-Baer and every countable family of fully invariant direct summand of M has a generalized countable join. Our results extend several existing results.

Journal of Mathematical Sciences, 1999

Communications in Algebra, 2016

Let R be an arbitrary ring with identity and M a right R-module with S = End R (M). Let F be a fully invariant submodule of M and I -1 (F) denotes the set {m ∈ M : Im ⊆ F} for any subset I of S. The module M is called F-Baer if I -1 (F) is a direct summand of M for every left ideal I of S. This work is devoted to investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F ⊕ N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.

2019

Let R be a non-necessarily commutative ring with unity 1 6= 0 and M a unitary left R-module. An R-module M is said to be generalized hopfian if every surjective R-endomorphism of M is superfluous. It is well known that any noetherian module is generalized hopfian but converse is not always true. For instance the Z-module Q of rational numbers is generalized hopfian but it is not noetherian. For a fixed ring R, we study R-modules for which every generalized hopfian module of σ[M ] is noetherian. Such modules are said to be generalized S-modules. In this paper, some properties and important characterizations of generalized S-modules are given. Keyword: hopfian module, generalized hopfian module, generalized S-module Introduction Let R be a non-necessarily commutative ring with unity 1 6= 0 and M a left module over R. Let M and N be two objects of R-Mod. We say that N is generated by M if there are a set Λ and a surjective homomorphism φ : M (Λ) → N . A submodule of N is said to be sub...

Algebra Colloquium, 2008

Let U be a submodule of a module M. We call U a strongly lifting submodule of M if whenever M/U=(A+U)/U ⊕ (B+U)/U, then M=P ⊕ Q such that P ≤ A, (A+U)/U=(P+U)/U and (B+U)/U=(Q+U)/U. This definition is a generalization of strongly lifting ideals defined by Nicholson and Zhou. In this paper, we investigate some properties of strongly lifting submodules and characterize U-semiregular and U-semiperfect modules by using strongly lifting submodules. Results are applied to characterize rings R satisfying that every (projective) left R-module M is τ (M)-semiperfect for some preradicals τ such as Rad , Z2 and δ.

Communications in Algebra, 2007