Ahmed Hamed, Achraf Malek Faculty of Sciences; Faculty of Sciences
Abstract : Let $R\subseteq T$ be an extension of a commutative ring and $S\subseteq R$ a multiplicative subset. We say that $(R, T)$ is an $S$-accr (a commutative ring $R$ is said to be $S$-accr if every ascending chain of residuals of the form $(I:B)\subseteq (I:B^2)\subseteq (I:B^3)\subseteq\cdots$ is $S$-stationary, where $I$ is an ideal of $R$ and $B$ is a finitely generated ideal of $R$) pair if every ring $A$ with $R\subseteq A\subseteq T$ satisfies $S$-accr. Using this concept, we give an $S$-version of several different known results.
Keywords : $S$-accr, $S$-Noetherian, pair of rings