S-Prime Right Submodules and an S-Version of Prime Avoidance
Alaa Abouhalaka
TL;DR
The paper addresses extending the classical prime and Noetherian concepts to the context of noncommutative rings through an $S$-structure, focusing on right $R$-modules. It introduces $S$-prime submodules and $S$-multiplication modules, establishes when $(P:_RM)$ being a right $S$-prime ideal corresponds to $P$ being $S$-prime, and proves an $S$-version of prime avoidance. It then develops $S$-finite and $S$-Noetherian notions for right modules and rings, proving a key result: for a multiplication finitely generated module $M$, if every $(N:_RM)$ is $S$-prime, then $M$ is $S$-Noetherian, with several stability results and examples. This work broadens the tools for noncommutative prime avoidance and Noetherian theory under $S$-structures, offering a framework for applications where the ring or module structure is filtered by an $m$-system.
Abstract
Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that in multiplication (S-multiplication) right R-module M, the ideal (P :_R M) is a right S-prime ideal of R if and only if P is an S-prime submodule of M. Moreover, we give an S-version of prime avoidance lemma. Furthermore, we define S-finite and S-Noetherian right modules following the definitions in [1]. We prove that a multiplication finitely generated right R-module M is S-Noetherian if (N :_R M) is an S-prime ideal of R, for all submodules N of M. In addition, we give some examples of right S-Noetherian rings.