On (ɱ,ɳ)-Strongly Fully Stably Banach Algebra Modules Related to an Ideal of Am ×ɳ

Abstract

The aim of this paper is introducing the concept of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal. Some properties of (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal have been studied and another characterizations have been given. The relationship of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal that states, a B- -module Ӽ is (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal , if and only if for any two ɱ-element sub-sets and of Ӽɳ, if , for each j = 1, …, ɱ, i = 1,…, ɳ and implies Ạɳ( ) Ạɳ( have been proved..

Weakly Nearly Semiprime Submodules and Some Related Concepts

“…(⟸) 0 ≠ 𝑟 𝑛 𝐿 ⊆ 𝐵 for 𝑟 ∈ 𝑅, and 𝐿 be a submodule of 𝐺, 𝑛 ∈ 𝑍 + . Since 𝐺 be multiplication Rmodule then 𝐿 = 𝐼𝐺 for some nonzero ideal 𝐼 of 𝑅, that is 0 ≠ 𝑟 𝑛 𝐼𝐺 ⊆ 𝐵, it follows that 0 ≠ 𝑟 𝑛 Recall that an R-module 𝐺 is a cancellation if 𝐼𝐺 = 𝐽𝐺 for any ideals 𝐼, 𝐽 of 𝑅, implies that 𝐼 = 𝐽 [12]. Proposition 3.11 Let 𝐺 be a multiplication faithful finitely generated projective R-module , and 𝐵 be a proper submodule of 𝐺.…”

“…But 𝐵 = [𝐵: 𝑅 𝐺]𝐺, hence [𝐵: 𝑅 𝐺]𝐺 = 𝐼𝐺. Since 𝐺 is a multiplication faithful finitely generated R-module then by [12,Prop. 3.1] 𝐺 is a cancellation, that is [𝐵: 𝑅 𝐺] = 𝐼 which is a WN-semiprime ideal of 𝑅.…”