SFT-stability and Krull dimension in power series rings over an almost pseudo-valuation domain (Q466967)

Let \(R\) be a commutative ring with identity. An ideal \(I\) is called an SFT-ideal if there exist a natural number \(k\) and a finitely generated ideal \(J\subseteq I\) such that \(a^k\in J\) for every \(a\in I\). A ring is called an SFT-ring if every ideal is an SFT-ideal. An integral domain \(R\) is called an almost pseudo-valuation domain (for short, APVD) if \(R\) is a quasi-local domain with maximal ideal \(M\) and there is a valuation overring in which \(M\) is a primary ideal. Among other results, it is shown that the power series ring \(R[[x_1,\dots,x_n]]\) over an APVD ring \(R\) with quotient field \(K\) and maximal ideal \(M\) 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 \((M:M)=\{x\in K: xM\subseteq M\}\).