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

Communications in Algebra, 2007

Bulletin of the Australian Mathematical Society

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1987

Introduction. Let R be a fixed (not necessarily commutative) ring. Throughout this nte, we are concerned with left R-modules M, A, H, Like in'Goldie [1], we shall use the following terminology. A non-zero submodule K of M is called essential in M (or M is an essential extension

Journal of Pure and Applied Algebra, 2013

A ring R is called a right weakly V-ring (briefly, a right WV-ring) if every simple right R-module is X-injective, where X is any cyclic right R-module with X R R R. In this note, we study the structure of right WV-rings R and show that, if R is not a right V-ring, then R has exactly three distinct ideals, 0 ⊂ J ⊂ R, where J is a nilpotent minimal right ideal of R such that R/J is a simple right V-domain. In this case, if we assume additionally that R J is finitely generated, then R is left Artinian and right uniserial with composition length 2. We also show that a strictly right WV-ring with Jacobson radical J is a Frobenius local ring if and only if the injective hull of J R is uniserial. Some other results are obtained in the connection with the Noetherian property of right WV-rings and related rings.

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.

ISRN Algebra, 2014

Çalışıcı and Türkmen called a module M generalized ⊕-supplemented if every submodule has a generalized supplement that is a direct summand of M. Motivated by this, it is natural to introduce another notion that we called generalized ⊕-radical supplemented modules as a proper generalization of generalized ⊕-supplemented modules. In this paper, we obtain various properties of generalized ⊕-radical supplemented modules. We show that the class of generalized ⊕-radical supplemented modules is closed under finite direct sums. We attain that over a Dedekind domain a module M is generalized ⊕-radical supplemented if and only if M/P(M) is generalized ⊕-radical supplemented. We completely determine the structure of these modules over left V-rings. Moreover, we characterize semiperfect rings via generalized ⊕-radical supplemented modules.

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 Mathematical Sciences, 2009

Let A be a ring that does not contain an infinite set of idempotents that are orthogonal modulo the ideal SI(AA). It is proved that all A-modules are I0-modules if and only if either A is a right semi-Artinian, right V-ring or A/ SI(AA) is an Artinian serial ring and the square of the Jacobson radical of A/ SI(AA) is equal to zero.

Journal of pure and applied algebra, 1992

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.