On a Generalization of Semisimple Modules

Abstract

Let R be a ring with identity. A module MR is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of MR which is a generalization of semisimple and second modules.We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals ≤4 and J (R) = 0 is r -semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring.