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