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 N of a module M , the semiprime radical of N in M is the direct limit of the ascending chain E M (N) ⊆ E M (E M (N)) ⊆ ⋯ of submodules of M , and we prove that N satisfies the semiprime radical formula if and only if its envelope E M (N) in M is semiprime. [ABSTRACT FROM AUTHOR]