Communications of the
Korean Mathematical Society CKMS

Let $Rsubseteq T$ be an extension of a commutative ring and $Ssubseteq R$ 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 $(I:B)subseteq (I:B^2)subseteq (I:B^3)subseteqcdots$ 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 $Rsubseteq Asubseteq T$ satisfies $S$-accr. Using this concept, we give an $S$-version of several different known results.

Keywords: $S$-accr, $S$-Noetherian, pair of rings

MSC numbers: Primary 13B, 13C, 13E05, 13E10