On $S$-Noetherian rings

Liu Zhongkui

Address. Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, People's Republic of China

E-mail: liuzk@nwnu.edu.cn

Abstract. Let $R$ be a commutative ring and $S\subseteq R$ a given multiplicative set. Let $(M,\leq)$ be a strictly ordered monoid satisfying the condition that $0\leq m$ for every $m\in M$. Then it is shown, under some additional conditions, that the generalized power series ring $[[R^{M,\leq}]]$ is $S$-Noetherian if and only if $R$ is $S$-Noetherian and $M$ is finitely generated.

AMSclassification. 16D40, 16S50.

Keywords. $S$-Noetherian ring, generalized power series ring, anti-Archimedean multiplicative set, $S$-finite ideal.