Prime–Extending Module and S-Prime Module

Abstract

In this paper, the notation of prime-extending module, is introduced and studied where an R-module M is Prime-extending if every nonzero proper submodule of M is essential in prime direct summand .Beside these, the notation of S-prime is given, where an R module is called S-prime if every proper direct summand is prime for this notation we proved: 1.Under the condition of S-prime module, M is Prime-extending if and only if M is extending .2. If M is semisimple module, then M is Prime-extending if and only if M is S-prime.