Generalizations of S-semiprime submodules

Abstract

Let [Formula: see text] be a commutative ring with non-zero identity, [Formula: see text] a multiplicatively closed subset of [Formula: see text] and [Formula: see text] a unital [Formula: see text]-module. In this paper, we introduce the concepts of [Formula: see text]-almost semiprime submodules and [Formula: see text]-weakly semiprime submodules as a generalization of [Formula: see text]-semiprime submodules. We investigate some basic properties of [Formula: see text]-almost semiprime and [Formula: see text]-weakly semiprime submodules and give some characterizations of them, especially, for (finitely generated faithful) multiplication modules.