On the endomorphism rings of max CS and min CS modules

Communications in Algebra, 2006

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.

Asian-European Journal of Mathematics, 2017

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

2013

We investigate the concept of s − CS modules. Basic properties of these modules are given. We prove that any right s − CS, right CF is artinian and we answer the F GF conjecture positively in case of R is s − CS. Some properties of s−injective rings are given.

Let R be a ring. A right R-module N is called an M-p-injective module if any homomorphism from an M-cyclic submodule of M to N can be extended to an endomorphism of M. Generalizing this notion, we investigate the class of M-rp-injective modules and M-lp-injective modules, and prove that for a finitely generated Kasch module M, if M is quasi-rp-injective, then there is a bijection between the class of maximal submodules of M and the class of minimal left right ideals of its endomorphism ring S. In this paper, we give some characterizations and properties of the structure of endomorphism rings of M-rp-injective modules and M-lp-injective modules and the relationships between them.

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.

Proceedings of The Edinburgh Mathematical Society, 1989

It is well known that a ring R is semiprime Artinian if and only if every right ideal is an injective right i?-module. In this paper we shall be concerned with the following general question: given a ring R all of whose right ideals have a certain property, what implications does this have for the ring R itself? In practice, it is not necessary to insist that all right ideals have the property, usually the maximal or essential right ideals will suffice. On the other hand, Osofsky proved that a ring R is semiprime Artinian if and only if every cyclic right /^-module is injective. This leads to the second general question: given a ring R all of whose cyclic right /^-modules have a certain property, what can one say about R itself?

Journal of Pure and Applied Algebra, 1995

If @ is any class of modules over a general ring R such that @ is closed under direct sums, quotients and submodules, then every module in @ is CS if and only if every module M in @ has a decomposition M = Ois, Mi, where each module Mi (i ~1) is either simple, or has length 2 and is X-injective for each module X in @. In consequence, necessary and sufficient conditions are given for a ring to have all its right singular modules CS. Rings whose finitely generated modules are CS are also studied. AMS ClassiJication: Primary 16D70; Secondary 16D50, 16D40, 16E50, 16P60 0022-4049/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved. SSDIOO22-4049(95)00084-V

Communications in Algebra, 2007

Pacific Journal of Mathematics