On Quasi-Small Prime Submodules

Abstract

Let be a commutative ring with identity , and be a unitary (left) R-module. A proper submodule of is said to be quasi- small prime submodule , if whenever with and , then either or . In this paper ,we give a comprehensive study of quasi- small prime submodules.

Quasi–essentially pseudo – prime modules

“…Recall that an R-module M is uniform if every nonzero submodule of M is an essential in M [4] . Since every essential submodule is a quasi-essential [6] then we have the following result.…”

“…Let R be a commutative ring with identity and M be a unitary R-module. An R-module M is called a prime if ann R M=ann R N for every non-zero submodule N of M [8].A proper submodule N of an R-module M is called Quasi-essential submodule in M if N Q≠(0) for each non-zero quasi-prime submodule Q of M [6] where a proper submodule Q of an R-module M is called a quasi-prime if r 1 r 2 m Q,m M ,r 1 ,r 2 R then either r 1 m Q or r 2 m Q [2]. And a proper submodule N of M is called a primary if r R , m M and rm N then m N , or r n [N:M] for some n Z + , where [N:M]={r R:rM⊆N}.…”

2-Prime Modules

“…Every Quasi-Dedekind module over a ring R is a 2-prime R-module. Proof: By [16] every Quasi-Dedekined is prime and hence it is 2-prime.…”