Semiprime submodules of a module and related concepts

In this paper, we consider the notion of semiprime submodules, which is a natural generation of semiprime ideals. The main purpose of this paper is to give local-global properties of semiprimes and semiprime radicals as well as results on finitely generated multiplication modules. Also, we prove that for every submodule [Formula: see text] of a module [Formula: see text], the semiprime radical of [Formula: see text] in [Formula: see text] is the direct limit of the ascending chain [Formula: see text] of submodules of [Formula: see text], and we prove that [Formula: see text] satisfies the semiprime radical formula if and only if its envelope [Formula: see text] in [Formula: see text] is semiprime.