Weakly Small Smiprime Submodules

Abstract

Let R be a commutative ring with an identity, and G be a unitary left R-module. A proper submodule H of an R-module G is called semiprime if whenever a ∈ R, y ∈ G, n ∈ Z + and any ∈ H, then ay ∈ H. We say that a properi submodule H of an R-module G is a weakly small semiprime, if whenever a ∈ R, y ∈ G, n∈Z +, (y) is small in G and 0 ≠ any ∈ H, implies ay ∈ H. Many basic properties and characterizations of this type of submodule are given.