On a Proper Subclass of Primeful Modules Which Contains the Class of Finitely Generated Modules Properly

Abstract

Let $R$ be a commutative ring with identity and $M$ a unital $R$-module. Moreover, let $PSpec(M)$ denote the primary-like spectrum of $M$ and $Spec(R/Ann(M))$ the prime spectrum of $R/Ann(M)$. We define an $R$-module $M$ to be a $\phi$-module, if $\phi:PSpec(M)\rightarrow Spec(R/Ann(M))$ given by$\phi(Q)=\sqrt{(Q:M)}/Ann(M)$ is a surjective map. The class of $\phi$-modules is a proper subclass of primeful modules, called $\psi$-modules here, and contains the class of finitely generated modules properly. Indeed, $\phi$ and $\psi$ are two sides of a commutative triangle of maps between spectrums. We show that if $R$ is an Artinian ring, then all $R$-modules are $\phi$-modules and the converse is true when $R$ is a Noetherian ring.