A characterization of artinian rings

Abstract

Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R-module is finitely embedded.

On certain classes of modules

“…of a module M) assuming certain classes of modules (associated to M) posses certain properties and viceversa . The results in the present paper are of a similar nature and are an outcome of results proved in [1], [2], [3], [4], [5] ; [6], [8] and [9] . In [1] among other results A. W .…”

Generalized injectivity and chain conditions

“…Next we will derive some consequences of these results. The first corollary is the main result of [9], which is turn is a generalization of [6] and [7].…”

A characterization of noetherian rings by cyclic modules

“…We are unable to derive this result from the above mentioned theorems of [3] and [7], respectively, or from the techniques of their proofs. The following result is an immediate consequence of the theorem and it gives a positive answer to (Q 2 ):…”

“…Dually, it is obtained in [7,Theorem 1.1] that a ring R is right artinian if and only if every cyclic right R-module is a direct sum of an injective module and a finitely cogenerated module. If we "combine" the assumptions on cyclic modules from these two results for a ring R, then R has right Krull dimension as established in [10] (see (1.1) …”