On generalized Krull power series rings
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Let $R$ be an integral domain. We prove that the power series ring $R[\![X]\!]$ is a Krull domain if and only if $R[\![X]\!]$ is a generalized Krull domain and $t$-$\dim R \leq 1$, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts ``Krull domain" and ``generalized Krull domain" are the same in power series rings, (2) there exists a non-$t$-SFT domain $R$ with $t$-$\dim R > 1$ such that $t$-$\dim R[\![X]\!] =1$.
Abstract
Let $R$ be an integral domain. We prove that the power series ring $R[\![X]\!]$ is a Krull domain if and only if $R[\![X]\!]$ is a generalized Krull domain and $t$-$\dim R \leq 1$, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts ``Krull domain" and ``generalized Krull domain" are the same in power series rings, (2) there exists a non-$t$-SFT domain $R$ with $t$-$\dim R > 1$ such that $t$-$\dim R[\![X]\!] =1$.
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