Some results on V-rings and weakly V-rings

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 XR≇RR. 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 RJ 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 JR is uniserial. Some other results are obtained in the connection with the Noetherian property of right WV-rings and related rings.