r-Submodules and sr-Submodules

Abstract

In this article, we introduce new classes of submodules called r-submodule and special r-submodule, which are two different generalizations of r-ideals. Let M be an R-module, where R is a commutative ring. We call a proper submodule N of M an r-submodule (resp., special r-submodule) if the condition am is an element of N with ann(M) (a) = 0(M) (resp., ann(R)(m) = 0) implies that m E N (resp., a is an element of (N :(R) M)) for each a is an element of R and m is an element of M. We also give various results and examples concerning r-submodules and special r-submodules.