A Note on⊕-Cofinitely Supplemented Modules

Let M be a right R-module. M is called -cofinitely -supplemented if every submodule N of M with M N finitely generated has a -supplement that is a direct summand of M. Any factor module of a distributive -cofinitely -supplemented module is -cofinitely -supplemented. Every cofinite direct summand of a -cofinitely -supplemented module with (D 3) is -cofinitely -supplemented. Arbitrary direct sum of -cofinitely -supplemented modules is -cofinitely -supplemented. For a module M with SSP, M is -cofinitely -supplemented if and only if every maximal submodule has a -supplement that is a direct summand of M. We show that a ring R is -semiperfect if and only if every free R-module is -cofinitely -supplemented.

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.

2018

In this paper, we introduce the concept of modules with the properties (RE) and (SRE), and we provide various properties of these modules. In particular, we prove that a semisimple module \(M\) is \(\operatorname{Rad}\)-supplementing if and only if \(M\) has the property (SRE). Moreover, we show that a ring \(R\) is a left V-ring if and only if every left \(R\)-module with the property (RE) is injective. Finally, we characterize the rings whose modules have the properties (RE) and (SRE).

Contemporary Mathematics, 2014

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 to δsupplemented modules and principally ⊕-supplemented modules. The module M is called principally ⊕-δ-supplemented if for any m ∈ M there exists a direct summand A of M such that M = mR + A and mR ∩ A is δ-small in A. We prove that some results of principally ⊕-supplemented modules can be extended to principally ⊕-δ-supplemented modules for this general settings. Several properties of these modules are given and it is shown that the class of principally ⊕-δ-supplemented modules lies strictly between classes of principally ⊕-supplemented modules and principally δ-supplemented modules. We investigate conditions which ensure that any factor modules, direct summands and direct sums of principally ⊕-δ-supplemented modules are also principally ⊕-δ-supplemented. We give a characterization of principally ⊕-δ-supplemented modules over a semisimple ring and a new characterization of principally δsemiperfect rings is obtained by using principally ⊕-δ-supplemented modules.

Hacettepe Journal of Mathematics and Statistics, 2016

Let I be an ideal of a ring R and let M be a left R-module. A submodule L of M is said to be δ-small in M provided M = L + X for any proper submodule X of M with M/X singular. An R-module M is called I-⊕-supplemented if for every submodule N of M , there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K is δ-small in K. In this paper, we investigate some properties of I-⊕-supplemented modules. We also compare I-⊕-supplemented modules with ⊕-supplemented modules. The structure of I-⊕-supplemented modules and ⊕-δ-supplemented modules over a Dedekind domain is completely determined.

Annals of the Alexandru Ioan Cuza University - Mathematics, 2015

In this paper, over an arbitrary ring we define the notion of weakly radical supplemented modules (or briefly wrs-module), which is adapted from Zöschinger's radical supplemented modules over a discrete valuation ring (DVR), and obtain the various properties of these modules. We prove that a wrs-module having a small radical is weakly supplemented. Moreover, we show that a ring R is left perfect if and only if every left R-module is wrs. Also, we prove that every wrs-module over a DVR is radical supplemented.

Bulletin of the iranian mathematical society

A module M is called H-co�nitely supplemented if for every co�nite submodule E (i.e. M=E is �nitely generated) of M there exists a direct summand D of M such that M = E+X holds if and only if M = D+X, for every submodule X of M. In this paper we study factors, direct summands and direct sums of H-co�nitely supplemented modules. Let M be an H-co�nitely supplemented module and let N � M be a submodule. Suppose that for every direct summand K of M, (N + K)=N lies above a direct summand of M=N. Then M=N is H-co�nitely supplemented. Let M be an H-co�nitely supplemented module. Let N be a direct summand of M. Suppose that for every direct summand K of M with M = N + K, N \ K is also a direct summand of M. Then N is H-co�nitely supplemented. Let M = M1�M2. If M1 is radical M2-projective (or M2 is radical M1-projective) and M1 and M2 are H-co�nitely supplemented, then M is H-co�nitely supplemented

Applied Categorical Structures, 2006

In this paper, it is shown that any non-M-cosingular ⊕-supplemented module M is (D 3) if and only if M has the summand intersection property. Let N ∈ σ [M ] be any module such that Z M (N) has a coclosure in N. Then we prove that N is (completely) ⊕-supplemented if and only if N = Z 2 M (N) ⊕ K for some submodule K of N such that Z 2 M (N) and K both are (completely) ⊕-supplemented.

Publications de l'Institut Math?matique (Belgrade), 2018

Let R be an associative ring with identity. We introduce the notion of semi-τ-supplemented modules, which is adapted from srs-modules, for a preradical τ on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for τ = Rad. We show that the class of semi-τ-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical τ on R-Mod, a module M is semi-τ-supplemented if and only if it is τ-supplemented. For τ = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.

2014

In this paper, we introduce principally ⊕-supplemented modules as a generalization of ⊕-supplemented modules and principally lifting modules. This class of modules is a strengthening of principally supplemented modules. We show that the class of principally ⊕-supplemented modules lies between classes of ⊕-supplemented modules and principally supplemented modules. We prove that some results of ⊕-supplemented modules and principally lifting modules can be extended to principally ⊕-supplemented modules for this general setting. We obtain some characterizations of principally semiperfect rings and von Neumann regular rings by using principally ⊕-supplemented modules.