Quasi-Radical Semiprime Submodules

Abstract

In this paper, we introduce the concept of a quasi-radical semi prime submodule. Throughout this work, we assume that is a commutative ring with identity and is a left unitary R- module. A proper submodule of is called a quasi-radical semi prime submodule (for short Q-rad-semiprime), if for , ,and then . Where is the intersection of all prime submodules of .

Closed (St-Closed) Compressible Modules and Closed (St-Closed) Retractable Modules

“…65, No. 4, pp: 2141-2149 A submodule P of M is said to be prime if P is a proper submodule U in M and whenever 𝑟𝑥 ∈ 𝑃 for all 𝑟 ∈ 𝑅 , 𝑥 ∈ 𝑀 up to either 𝑥 ∈ 𝑃 or 𝑟 ∈ [𝑃: 𝑀], where [𝑃: 𝑀] = {𝑟 ∈ 𝑅: 𝑟𝑀 ⊆ 𝑃}, [2,12]. A submodule U of an R-module Y is termed St-closed (briefly 𝑁 ≤ 𝑆𝑡𝑐 𝑀), if N has no proper semi-essential extension in M, i.e., if there exist a submodule K of M such that 𝑁 ≤ 𝑆𝑡𝑐 𝐾 ≤ 𝑀 , then 𝑁 = 𝐾, [3].…”

“…Recall that an R-module M is called a fully essential if every submodule in M is essential, [2]. Recall that an R-module M is called semi-essentially compressible if 𝑀 can be embedded in every of it is a non-zero submodule of 𝑀. Equivalently, 𝑀 is compressible if there exists a non-zero monomorphism 𝑓: 𝑀 ⟶ 𝑁 whenever 0 ≠ 𝑁 ≤ 𝑠.𝑒 𝑀, [13].…”

“…The direct sum of a closed prime module need not be closed prime. Consider the following instances let 𝑍 6 ≃ 𝑍 3 ⨁𝑍 2 as 𝑍 −module.𝑍 3 and 𝑍 2 are closed prime submodules see (4) , however 𝑍 6 is not closed prime see(2) .…”

S-Essentially Compressible Modulesand S-Essentially Retractable Modules

“…Recall that A ring 𝑅 is said to be regular (von-numann) if for each 𝑎 ∈ 𝑅 there exists an element 𝑡 ∈ 𝑅 such that 𝑎 = 𝑎𝑡𝑎 (if 𝑅 is a commutative ring, then 𝑎 = 𝑎 2 𝑡 ), [9]. Proposition 4.10: Let 𝑀 be an indecomposable and an s-essentially retractable R-module.…”