The effect of $S$-accr on intermediate rings between certain pairs of rings

Authors : Subramanian VISWESWARAN

Pages : 101-128

Doi:10.24330/ieja.1096895

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Publication Date : 2022-07-16

Article Type : Research Paper

Abstract :The rings considered in this article are commutative with identity and the modules are assumed to be unitary. If $R$ is a subring of a ring $T$, then it is assumed that $R$ contains the identity element of $T$. Let $S$ be a multiplicatively closed subset insert ignore into journalissuearticles values(m.c. subset); of a ring $R$. In this paper, we consider the concept of $S$-accr, the generalization by Hamed and Hizem of the notion of insert ignore into journalissuearticles values(accr); in module theory given by Lu. We say that $R$ satisfies insert ignore into journalissuearticles values(accr); if the increasing sequence of residuals of the form $insert ignore into journalissuearticles values(I:_{R}B);\subseteq insert ignore into journalissuearticles values(I:_{R}B^{2});\subseteq insert ignore into journalissuearticles values(I:_{R}B^{3});\subseteq \cdots$ is stationary for any ideal $I$ of $R$ and for any finitely generated ideal $B$ of $R$. Focusing on certain pairs of rings $R\subseteq T$, the aim of this paper is to study whether $S$-accr on each intermediate ring $A$ between $R$ and $T$ for a suitable m.c. subset $S$ of $A$ insert ignore into journalissuearticles values(depending on $A$); implies that $A$ satisfies insert ignore into journalissuearticles values(accr); for each such $A$.
Keywords : Accr pair, S accr pair, Laskerian pair, strongly Laskerian pair, S Noetherian pair