On Bass Modules and Semi-V-Modules

Abstract

Let R be a commutative ring with identity. An $R$-module $M$ is called a semi-V-module if every nonzero homomorphic image of $M$ contains a nonzero V-submodule. An $R$-module $M$ is called a Bass module if every nonzero module in $\sigma [M]$ has a maximal submodule. It is shown that an $R$-module $M$ is a semi-V-module if and only if every nonzero cyclic submodule of $M$ is a Bass module if and only if $R/Ann_R(x)$ is a Bass ring for every nonzero element $x \in M$. Among other results, we characterize the class of rings $R$ for which every semi-V-module is a V-module and the class of rings $R$ for which every semi-V-module is a semisimple module. Also, we characterize the rings $R$ for which the class of semi-V-modules (Bass modules) is closed under direct products.