Some results on V-rings and weakly V-rings

Abstract

a b s t r a c tA 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.

Absolutelys-Pure Modules and Neat-Flat Modules

“…Recently, there is a significant interest to some classes of modules that are defined via simple modules, (see, [1], [4], [5], [6], [9], [20], [21], [25], [37], [38]). …”