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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.

On modules satisfying the descending chain condition on r -submodules