SFT-stability and Krull dimension in power series rings over an almost pseudo-valuation domain

Abstract

Let $$R$$ be an APVD with maximal ideal $$M$$ . We show that the power series ring $$R[[x_1,\ldots ,x_n]]$$ is an SFT-ring if and only if the integral closure of $$R$$ is an SFT-ring if and only if ( $$R$$ is an SFT-ring and $$M$$ is a Noether strongly primary ideal of $$(M:M)$$ ). We deduce that if $$R$$ is an $$m$$ -dimensional APVD that is a residually *-domain, then dim $$R[[x_1,\ldots ,x_n]]\,=\,nm+1$$ or $$nm+n$$ .