Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
References

A note on generalizations of semisimple modules

Profile image of Burcu Nişancı TürkmenBurcu Nişancı Türkmen

Abstract

A left module M over an arbitrary ring is called an RD-module (or an RS-module) if every submodule N of M with Rad(M) ⊆ N is a direct summand of (a supplement in, respectively) M. In this paper, we investigate the various properties of RD-modules and RS-modules. We prove that M is an RD-module if and only if M = Rad(M) ⊕ X, where X is semisimple. We show that a finitely generated RS-module is semisimple. This gives us the characterization of semisimple rings in terms of RS-modules. We completely determine the structure of these modules over Dedekind domains.

Related papers

A note on modules
Bhavanari Satyanarayana

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

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

View PDFchevron_right
MODULES WITH THE PROPERTY Rad g
Burcu Nişancı Türkmen

In this article, we introduce modules with the property Rad g and provide various properties of this module. Firstly, we prove that every semisimple module has the property Rad g. We also indicate that the class of modules with the property Rad g is closed under finite direct sum and direct summands. Moreover, we define the concepts of g-(radical) cover. We show that factor modules of a module with the property Rad g have the property Rad g under special condition. Additionally, we characterize semisimple commutative rings via modules with the property Rad g .

View PDFchevron_right
On Dedekind Modules
Bülent Saraç

Communications in Algebra, 2007

View PDFchevron_right
Semigroups of Modules: A Survey
Roger Wiegand
View PDFchevron_right
Modules which have a rad-supplement that is a direct summand in every extension
Burcu Nişancı Türkmen

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).

View PDFchevron_right
A GENERALIZATION OF SEMIPERFECT MODULES
Burcu Nişancı Türkmen

UDC 512.5 A module M is called radical semiperfect if M N has a projective cover whenever Rad(M) ✓ N ✓ M. We study various properties of these modules. It is proved that every left R-module is radical semiperfect if and only if R is left perfect. Moreover, radical lifting modules are defined as a generalization of lifting modules.

View PDFchevron_right
On a class of semicommutative modules
Nazim Tahir

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.

View PDFchevron_right
On Semicommutative Modules and Rings
Abdullah Harmanci

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.

View PDFchevron_right
On some classes of modules and characterization of rings
Carlos Signoret

Communications in Algebra, 2017

An abstract class of R-Mod is termed Additive Class if it is closed under taking submodules, homomorphic images and finite direct sums. In this paper we study the lattice properties of R-ad, the big lattice of all additive classes in R-mod for R any ring. Also, we study the lattice properties of R-ad when R is a Noetherian or Semisimple ring. We describe a characterization of some rings through the additive classes and their properties. We study some properties of additive classes related with the associated linear filters.

View PDFchevron_right
A generalization of modules with the property (P * )
Burcu Nişancı Türkmen

I.A-Khazzi and P.F. Smith called a module M have the property (P *) if every submodule N of M there exists a direct summand K of M such that K ≤ N and N K ⊆ Rad(M K). Motivated by this, it is natural to introduce another notion that we called modules that have the properties (GP *) and (N-GP *) as proper generalizations of modules that have the property (P *). In this paper we obtain various properties of modules that have properties (GP *) and (N-GP *). We show that the class of modules for which every direct summand is a fully invariant submodule that have the property (GP *) is closed under finite direct sums. We completely determine the structure of these modules over generalized f-semiperfect rings. 2010 Mathematics Subject Classification. 16D10, 16N80. Key words and phrases. Generalized f-semiperfect ring, the properties (P *), (GP *) and (N-GP *).

View PDFchevron_right

References (8)

  1. Alizade R., Bilhan G., Smith P. F., Modules whose maximal submodules have supplements, Comm. Algebra 29 (2001), no. 6, 2389-2405.
  2. Büyükaşık E., Pusat-Yılmaz D., Modules whose maximal submodules are supplements, Hacet. J. Math. Stat. 39 (2010), no. 4, 477-487.
  3. Büyükaşık E., Türkmen E., Strongly radical supplemented modules, Ukrainian Math. J. 63 (2012), no. 8, 1306-1313.
  4. Lomp C., On semilocal modules and rings, Comm. Algebra (1999), no. 4, 1921-1935.
  5. Nebiyev C., Pancar A., On supplement submodules, Ukrainian Math. J. 65 (2013), no. 7, 1071-1078.
  6. Türkmen B. N., Pancar A., Generalizations of ⊕-supplemented modules, Ukrainian Math. J. 65 (2013), no. 4, 612-622.
  7. Türkmen B. N., Türkmen E., On a generalization of weakly supplemented modules, An. S ¸tiin. Univ. Al. I. Cuza Din Iaşi. Mat. (N.S.) 63 (2017), no. 2, 441-448.
  8. Wisbauer R., Foundations of Module and Ring Theory, A handbook for study and research, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, 1991.

Related papers

A decomposition theorem for -semisimple rings
Hai Dinh

Journal of Pure and Applied Algebra, 2004

A module M is said to satisfy the condition (˝ *) if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisÿes (˝ *) are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right˝ *-semisimple rings. Right˝ *-semisimple rings are right artinian. However, in general, a right˝ *-semisimple rings need not be left˝ *-semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left˝ *-semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.

View PDFchevron_right
Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem
Asghar Daneshvar

Communications in Algebra

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module R M is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) virtually semisimple if R R is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring R is left completely virtually semisimple if and only if R ∼ = k i=1 M ni (D i) where k, n 1 , ..., n k ∈ N and each D i is a principal left ideal domain. Moreover, the integers k, n 1 , ..., n k and the principal left ideal domains D 1 , ..., D k are uniquely determined (up to isomorphism) by R.

View PDFchevron_right
Semi-Endosimple Modules and Some Applications
Mohammad Baziar

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.

View PDFchevron_right
A Generalization of Semiregular and Semiperfect Modules
Pınar Aydoğdu

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.

View PDFchevron_right
Semidistributive modules
A. Tuganbaev

Journal of Mathematical Sciences, 1999

View PDFchevron_right
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved