Almost strong finite type rings and Krull dimension of power series ring extensions from sequences (Q6608207)

Let \(\mathbb{N}_0\) (resp. \(\mathbb{N}\)) be the set of nonnegative (resp. positive) integers. Let \(\lambda:\mathbb{N}_0\longrightarrow\mathbb{N}\) be a function satisfying \(\lambda(0)=1\) and \(\lambda(i)\lambda(j)\) divides \(\lambda(i+j)\) in \(\mathbb{N}\) for each \(i\) and \(j\). Denote \(\alpha_{ij}=\frac{\lambda(i+j)},{\lambda(i)\lambda(j)}\in\mathbb{N}\). Let \(R\) be a commutative ring with identity and \(R[[X]]\) the set of formal power series with coefficients in \(R\). Define a multiplication \(*_{\lambda}\) in \(R[[X]]\) by \(\displaystyle(\sum_{i:0}^{\infty}a_iX^i)*_{\lambda}(\sum_{j:0}^{\infty}a_jX^j) =\sum_{k:0}^{\infty}(\sum_{i+j=k}\alpha_{ij}a_ib_j)X^)\). With the usual addition and this new multiplication, \(R[[X]]\) is a commutative ring that contains \(R\) as a subring. We denote this ring by \(R^{\lambda}[[X]]\). An ideal \(I\) of \(R\) is called an ASFT ideal if there exist a finitely generated ideal \(J\) of \(R\) with \(J\subseteq I\) and a positive integer \(k\) such that for each \(a\in I\), \(na^k\in J\) for some positive integer \(n\) (depending on \(a\)). The ring \(R\) is called an ASFT ring if every ideal of \(R\) is an ASFT ideal. For an infinite cardinal number \(\alpha\), we say that \(\dim(R)\geq\alpha\) if there exists a chain of prime ideals in \(R\) with lenght \(\geq\alpha\). In the paper under review, the authors first study the basic properties of ASFT ideals and ASFT rings. Then they show that if \(R\) is a non-ASFT ring, then \(\dim(R^{\lambda}[[X]])\geq 2^{\aleph_1}\).