On modules satisfying the descending chain condition on r-submodules

Abstract

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we introduce the concept of r-Artinian modules which is a new generalization of Artinian modules. An R-module M is called an r-Artinian module if M satisfies the descending chain condition on r-submodules. Also, we call the ring R to be an r-Artinian ring if R is an r-Artinian R-module. We prove that an R-module M is an r-Artinian module if and only if its total quotient module is an Artinian module. In particular, we observe that r-Artinian modules generalize S-Artinian modules, for some particular multiplicatively closed subsets S of R. Also, we extend many properties of Artinian modules to r-Artinian modules. Finally, we use the idealization construction to give non-trivial examples of r-Artinian rings that are not Artinian.