Some Results on Quasi-Semiprime Submodules

Abstract

Let R be a commutative ring with unity and let B be a submodule of a non-zero left R-module D, B is called semiprime if whenever a k y ∈ B, a ∈ R, y ∈ D, k ∈ Z + implies a y ∈ B. We say that a proper submodule B of an R-module D is a quasi-semiprime submodule if whenever a k b y ∈ B, where a, b ∈ R, y ∈ D, k ∈ Z + implies that a b y ∈ B. Equivalently, a proper submodule B of an R-module D is said to be a quasi-semiprime submodule if and only if [B ∶ (y)] is a semiprime ideal of R for each y ∈ D. We give many results of this type of submodules.