Small Prime Modules and Small Prime Submodules

Abstract

Let R be a commutative ring with identity, and M be a unital (left) R-module. In this paper we introduce and study the concepts: small prime submodules and small prime modules as generalizations of prime submodules and prime modules.Among the results that we obtain is the following: An R-module M is small prime if and only if the R-module R/annM is cogenerated by every non-trivial small submodule of M.

SS-Coprime Modules

“…Since M is SS-coprime , then M is S-coprime by Remarks and Examples 2. It is clear that every prime module is a small prime module , and if M is a small prime module, then annM is a prime ideal [8] Theorem 2.17 :…”

“…See Remarks and Examples 2.2 (8) Let M be an R-module , we say that M is small retractable if Hom(M,N)≠0 for each N<<M .…”