Amalgamated algebras along an ideal defined by accr-like conditions

Abstract

Let $R$ be a commutative ring with unity and $M$ an $R$-module. We say that $M$ satisfies accr if the ascending chain of submodules of the form $N: B\subseteq N: B^2\subseteq\cdots$ terminates for every submodule $N$ of $M$ and every finitely generated ideal $B$ of $R.$ The ring $R$ satisfies the accr condition if the $R$-module $R$ does. In this paper, we give a necessary and sufficient condition for amalgamated algebras $A\bowtie^fJ$ (respectively, the trivial extension of $R$ by $M,$ $R(+)M$) to be an accr ring. In the second part of this paper, we introduce the concept of uniformly $S$-accr modules and study its properties. Let $S$ be a multiplicative subset of $R.$ We say that $M$ satisfies uniformly $S$-accr if there exists an $s\in S$ such that for every submodule $N$ of $M$ and every finitely generated ideal $B$ of $R$ the ascending sequence of the form $N:B\subseteq N:B^2\subseteq \cdots$ is stationary with respective to $s.$ We study basic results of uniformly $S$-accr ring. Also, we give an $S$-version of several well-known result about accr rings. The concept of Nagata's idealization $R(+)M$ and the amalgamated algebras $A\bowtie^fJ$ is studied.