r -Submodules and sr -Submodules

Abstract

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