ON SEMIPRIME MODULES WITH CHAIN CONDITIONS

Abstract

Let R be an arbitrary ring, M a rightR-module and S = EndR(M), the endomorphism ring of M. A proper fully invariant submodule X of M is called a prime submodule of M if for any ideal I of S and any fully invariant submodule U of M, if I(U) ⊂ X, then either I(M) ⊂ X or U ⊂ X. A submodule X of M is called a semiprime submodule of M if it is an intersection of prime submodules. The module M is called a prime module if 0 is a prime submodule of M, and semiprime if 0 is a semiprime submodule of M. In this paper, we present some results on the classes of semiprime modules with chain conditions.