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On $S$-Noetherian rings

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*
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.