A question about saturated chains of primes in Serre conjecture rings
Let \(R, S\) be commutative rings with identity of finite dimensional. A domain \(R\) is said to be an \(S\)-domain if for each height one prime ideal \(p\) of \(R\) the extended prime \(p[X]\) is also height one. Let \(R\) be a ring, we say that \(R\) is a strong \(S\)-ring if, \(R/p\) is an \(S\)-domain for each prime ideal \(p\) of \(R\). The Serre conjecture ring \(R\langle X \rangle\) is defined as follows: \[ R\langle X \rangle:= \{ f/g \mid f, g \in R[X] \text{ and } g \text{ is monic} \}. \] By induction, one can define the Serre coots, conjecture ring in \(n\) indeterminates as \(R\langle n \rangle=R\langle X_1, \ldots, X_n \rangle:= R\langle X_1, \ldots, X_{n-1} \rangle \langle X_n \rangle\). The authors of the paper, give a positive answer to the following question with some extra assumptions. {Question}. Is \(R \langle 2 \rangle\) strong \(S\) when \(R \langle 1 \rangle\) is strong \(S\)?
scientific article; zbMATH DE number 6983435
scientific article