ON SEMIPRIME MODULES WITH CHAIN CONDITIONS

Abstract

Let R be an arbitrary ring, M a right R-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.