A note on maximal non-Noetherian subrings of a domain
A domain \(R\) is said to be an S(=Seidenberg)-domain if each height one prime ideal \(P \subset R\) extends to a height one prime ideal \(P[X] \subset R[X]\), where \(R[X]\) denotes the polynomial ring in one variable \(X\) over \(R\). A domain \(R\) is called a strong S-domain if \(R/P\) is an S-domain for each prime ideal \(P \subset R\). A domain \(R\) is called stably strong S-domain provided the polynominal ring \(R[X_1,\ldots,X_n]\) is a strong S-domain for any \(n \geq 1\). In the paper it is shown: If \(R\) is a maximal non-Noetherian subring of \(S\) then \(R\) is a stably strong S-domain and that \(R\) is universally catenerian if and only if \(S\) is universally catenerian. The results lead to new examples of stably strong S-domains and universally catenarian domains. The relationship with \(n\)-dimensional pairs and residually Mori pairs is also investigated.
scientific article; zbMATH DE number 6010043
scientific article