Journal of the
Korean Mathematical Society JKMS

In \cite{Hamed1}, the authors generalize the concept of the class group of an integral domain $D$ ($Cl_t(D)$) by introducing the notion of the $S$-class group of an integral domain where $S$ is a multiplicative subset of $D.$ The $S$-class group of $D,$ $S$-$Cl_t(D),$ is the group of fractional $t$-invertible $t$-ideals of $D$ under the $t$-multiplication modulo its subgroup of $S$-principal $t$-invertible $t$-ideals of $D.$ In this paper we study when $S$-$Cl_t(D)$ $\simeq$ $S$-$Cl_t(D_T),$ where $T$ is a multiplicative subset generated by prime elements of $D.$ We show that if $D$ is a Mori domain, $T$ a multiplicative subset generated by prime elements of $D$ and $S$ a multiplicative subset of $D,$ then the natural homomorphism $S$-$Cl_t(D) \rightarrow S$-$Cl_t(D_T)$ is an isomorphism. In particular, we give an $S$-version of Nagata's Theorem \cite{Nagata}: Let $D$ be a Krull domain, $T$ a multiplicative subset generated by prime elements of $D$ and $S$ another multiplicative subset of $D.$ If $D_T$ is an $S$-factorial domain, then $D$ is an $S$-factorial domain.

Keywords: $S$-Class group, Mori domain, divisorail ideal, $S$-principal ideal

MSC numbers: Primary 13C20, 13F05, 13A15, 13G05