Modules Whose Certain Submodules Are Prime

Abstract

Modules in which every proper submodule (resp. proper nonzero submod- ule) is prime (called fully prime (almost fully prime)) and with some other related notions are fully investigated. It is shown that over a commutative ring R ,a nR- module M is fully prime (fully semiprime) if and only if M is a homogeneous semisim- ple (co-semisimple) module. This in particular shows that a f.g. R-module M is co-semisimple if and only if R Ann (M) is a regular (von-Neumann) ring. Modules in which nonzero direct summands are prime are also characterized. When R is a one- dimensional Noetherian domain we determine all modules in which the zero submodule is the only prime (semiprime) submodule. Finally, we observe that R is a Max-ring if and only if every R-module contains a prime (semiprime) submodule.