A Relationship Between 2-Primal Modules and Modules That Satisfy the Radical Formula

Abstract

Abstract. The coincidence of the set of all nilpotent elements of a ring with its prime radical has a module analogue which occurs when the zero submodule satisfies the radical formula. A ring R is 2-primal if the set of all nilpotent elements of R coincides with its prime radical. This fact motivates our study in this paper, namely; to compare 2-primal submodules and submodules that satisfy the radical formula. A demonstration of the importance of 2-primal modules in bridging the gap between modules over commutative rings and modules over noncommutative rings is done and new examples of rings and modules that satisfy the radical formula are also given.Mathematics Subject Classification (2010): 16S90, 16N80, 16N60

A complete radical formula and 2-primal modules

“…Proof: Follows from the fact that for any ring R, Corollary 4.4 allows us to paraphrase Question 2.1 posed in paper [24] as: Question 4.1. Is there a prime (resp.…”

δ-small submodule and prime modules

“…Any proper submodule 𝐴 of ℳ is called prime submodule of ℳ if for each ideals 𝐽 of 𝑅 and 𝐴 1 ≤ ℳ such that 𝐽𝐴 1 ⊆ 𝐴, so 𝐴 1 ⊆ 𝐴 or 𝐽ℳ ⊆ 𝐴. A definition of prime module [Ssevviiri, 2011]. Any submodule 𝐴 of ℳ is called a completely prime submodule if for every 𝑟 ∈ 𝑅, 𝑚 ∈ ℳ such that 𝑟𝑚 ∈ 𝐴, so 𝑚 ∈ 𝐴 module in [Ssevviiri, 2013].…”

“…[Ssevviiri, 2011] An 𝑅-module ℳ is called prime if 𝑟𝑅𝑚 = 0, so 𝑟𝑚 = 0 or 𝑎 = 0 ∀𝑟 ∈ 𝑅, 𝑚 ∈ ℳ [Ssevviiri, 2013]. An 𝑅-module ℳ is called completely prime module (𝑐-𝑝𝑟-module) if 𝑟𝑚 = 0, so 𝑟 ∈ 𝑎𝑛𝑛 𝑅 (ℳ) or 𝑚 = 0 ∀𝑟 ∈ 𝑅, 𝑚 ∈ ℳ.…”

Uniformly Primal Submodule over Noncommutative Ring

“…e concept of uniformly primal submodules has been introduced and studied by Dauns in [5]. A submodule N of M is called a uniformly primal submodule provided that the set adj(N) � x ∈ R|mRr ⊆ N for some m ∈ M { } is uniformly not prime to N, where the subset B of R is uniformly not right prime to N if there exists an element s ∈ M − N with sRB ⊆ N. In particular, a number of papers concerning primal submodules have been studied by various authors (see, for example, [6][7][8][9][10]). In Section 2, we give some basic results about uniformly primal submodules and show that N 1 , N 2 , .…”