Generalized S-modules (original) (raw)

A generalization of supplemented modules

Algebra and Discrete Mathematics

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a class of modules which is an analogous of δ-supplemented modules defined by Kosan. The module M is called principally δ-supplemented, for all m ∈ M there exists a submodule A of M with M = mR+A and (mR)∩A δsmall in A. We prove that some results of δ-supplemented modules can be extended to principally δ-supplemented modules for this general settings. We supply some examples showing that there are principally δ-supplemented modules but not δ-supplemented. We also introduce principally δ-semiperfect modules as a generalization of δ-semiperfect modules and investigate their properties.

A note on modules

Proceedings of The Japan Academy Series A-mathematical Sciences, 1987

Introduction. Let R be a fixed (not necessarily commutative) ring.

S-NOETHERIAN RINGS, MODULES AND THEIR GENERALIZATIONS

Surveys in Mathematics and its Applications 18 (2023), 163 – 182, https://www.utgjiu.ro/math/sma, 2023

Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented.

On a class of semicommutative modules

Proceedings of The Indian Academy of Sciences-mathematical Sciences, 2009

Let R be a ring with identity, M a right R-module and S = End R (M). In this note, we introduce S-semicommutative, S-Baer, S-q.-Baer and S-p.q.-Baer modules. We study the relations between these classes of modules. Also we prove if M is an S-semicommutative module, then M is an S-p.q.-Baer module if and only if M[x] is an S[x]-p.q.-Baer module, M is an S-Baer module if and only if M[x] is an S[x]-Baer module, M is an S-q.-Baer module if and only if M[x] is an S[x]-q.-Baer module.

Semi-Endosimple Modules and Some Applications

2009

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module M R we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As consequences of these results, new characterizations of V-rings, right Noetherian V-rings and strongly semiprime rings are obtained. As such, a hereditary left Noetherian ring R is a finite direct sum of simple Noetherian right V-rings if and only if all finitely generated right R-modules are semi-endosimple.

A Generalization of Rickart Modules

Bulletin of the Belgian Mathematical Society - Simon Stevin

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). In this paper we introduce π-Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart modules. The module M is called π-Rickart if for any f ∈ S, there exist e 2 = e ∈ S and a positive integer n such that rM (f n) = eM. We prove that several results of Rickart modules can be extended to π-Rickart modules for this general settings, and investigate relations between a π-Rickart module and its endomorphism ring.

Rings over which all modules are I 0-modules. II

Journal of Mathematical Sciences, 2009

All right R-modules are I0-modules if and only if either R is a right SV-ring or R/I (2) (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I (2) (R) is equal to zero. All rings are assumed to be associative and with nonzero identity element. Expressions such as "an Artinian ring" mean that the corresponding right and left conditions hold. A submodule X of the module M is said to be superfluous in M if X + Y = M for every proper submodule P of the module M. Following [9], we call a module M an I 0-module if every nonsuperfluous submodule of M contains a nonzero direct summand of the module M. It is clear that I 0-modules are weakly regular modules, considered in [1-3, 8]; a module M is said to be weakly regular if every submodule of M that is not contained in the Jacobson radical of M contains a nonzero direct summand of M. Weakly regular modules are studied in [1-3; 6; 8; 9; 11, Chap. 3; 12-14], and other papers. A ring R is called a right generalized SV-ring if every right R-module is weakly regular. It is clear that A is a right generalized SV-ring provided each right A-module is an I 0-module. In addition, it follows from the presented paper that the Jacobson radical of any right module over a right generalized SV-ring is superfluous; therefore, every right module over a right generalized SV-ring is an I 0-module. The aim of the paper is the study of generalized right SV-rings. The main result of the present paper is Theorem 1.

Some properties of purely simple Kronecker modules, I

Journal of Pure and Applied Algebra, 1983

Let K be an algebraically closed field. A K2-system is a pair of K-vector spaces (V, W) together with a K-bilinear map from K' x V to W. The category of systems is equivalent to the category of right modules over some K-algebra, R. Most of the concepts in the theory of modules over the polynomial ring K[<] have analogues in Mod-R. Unlike the purely simple K(C]-modules, which are easily described, purely simple R-modules are quite complex. If M is a purely simple R-module of finite rank n then any submodule of M of rank less than n is finite-dimensional. The following corollaries are derived from this fact: 1, Every non-zero endomorphism of M is manic. 2. Every torsion-free quotient of M is purely simple. 3. An ascending union of purely simple R-modules of increasing rank is not purely simple. It is also shown that a large class of torsion-free rank one modules can occur as the quotient of a purely simple system of rank n, n any positive integer. Moreover. starting from a purely simple system another purely simple module M' of the same rank is constructed and M' is shown to be both a submodule of M and a submodule of a rank 1 torsion-free system. Since the category of right R-modules is a full subcategory of right S-modules, where S is any finite-dimensional hereditary algebra of tame type, the paper provides a way of constructing infinite-dimensional indecomposable S-modules.

A Generalization of Semiregular and Semiperfect Modules

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.