Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring with identity and $S$ be a multiplicatively closed subset of $R$. In this article we introduce a new class of ring, called $S$-multiplication rings which are $S$-versions of multiplication rings. An $R$-module $M$ is said to be $S$-multiplication if for each submodule $N$ of $M$, $sN\subseteq JM\subseteq N$ for some $s\in S$ and ideal $J$ of $R$ (see for instance  [4, Definition 1]). An ideal $I$ of $R$ is called $S$-multiplication if $I$ is an $S$-multiplication $R$-module. A commutative ring $R$ is called an $S$-multiplication ring if each ideal of $R$ is $S$-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and $S$-$PIR$. Moreover, we generalize some properties of multiplication rings to $S$-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

Keywords: Multiplication ideals, $S$-multiplication ideal, multiplication rings, almost multiplication rings, arithmetical rings, $S$-multiplication rings, $S$-arithmetical rings, trivial ring extension, amalgamated algebra along an ideal, amalgamated duplication

MSC numbers: 13A15, 13F10