PID pairs of rings and maximal non-PID subrings
Let \(R\subset S\) be a domain extension. \((R,S)\) is a principal ideal domain (shortly PID) pair if every ring \(T\), \(R\subset T\subset S\) is a PID. If \(R\) is not a field then it shows that the only PID pairs are those such that \(R\) is a PID. If \(R\) is a field then \((R,S)\) is a PID pair iff \(S\) is algebraic over \(R\). If \(R\) and \(S\) share a nonzero ideal, then \(R\) is a maximal non-PID subring of \(S\) iff \(S\) is a PID, \(R\) is not a PID and \(R\subset S\) is a minimal extension, that is it has no intermediate rings. Suppose that \(R\) is a Dedekind domain with infinitely many maximal ideals. Then \(R\) is a maximal non-PID subring of \(S\) iff \(R\) has a cyclic divisor class group generated by the class of any maximal ideal of \(R\).
principal ideal domain, Dedekind domain, Prüfer domain
scientific article; zbMATH DE number 5944182
scientific article