Generalized injectivity and chain conditions

Abstract

Relationships between injectivity or generalized injectivity and chain conditions on a module category have been studied by several authors. A well-known theorem of Osofsky [14, 15] asserts that a ring all of whose cyclic right modules are injective is semisimple Artinian. Osofsky's proofs in [14, 15] essentially used homological properties of injective modules, and, later, her arguments were applied by other authors in their studies of rings for which cyclic right modules are quasi-injective, continuous or quasi-continuous (see e. g. [1, 10, 12]). Following [5] (cf. [4]), a module M is called a CS-module if every submodule of M is essential in a direct summand of M. In the recent paper [17], B. L. Osofsky and P. F. Smith have proved a very general theorem on cyclic completely CS-modules from which many known results in this area follow rather easily. In another direction, it was proved in [8] that a finitely generated quasi-injective module with ACC (respectively DCC) on essential submodules is Noetherian (respectively Artinian). This result was also extended to CS-modules in [3, 16], and weak CS-modules in [19].