On the Dual of Weakly Prime and Semiprime Modules

Abstract

The weakly second modules (the dual of weakly prime modules) was introduced in [6]. In this paper we introduce and study the semisecond and strongly second modules. Let R be a ring and M be an R-module. We show that M is semisecond if and only if MI = MI² for any ideal I of R. It is shown that every sum of the second submodules of M is a semisecond submodule of M. Also if M is an Artinian module, then M has only a finite number of maximal semisecond submodules. We prove that every strongly second submodule of M is second and every minimal submodule of M is strongly second. If every nonzero submodule of M is (weakly) second, then M is called fully (weakly) second. It is shown that if R is a commutative ring, then M is fully second if and only if M is fully weakly second, if and only if M is a homogeneous semisimple module.

Key words and phrases. weakly second modules, semisecond modules, weakly prime modules.

2000 Mathematics Subject Classification. 16D10, 16D80, 16N60