Maximal non-prime ideally equal subrings of a commutative ring (Q1623075)

Let \(R\) be a proper subring of a commutative ring \(S\). The ring \(R\) is said to be maximal non-prime ideally equal subring of \(S\), if \(\mathrm{Spec}(R)\neq\mathrm{Spec}(S)\), whereas \(\mathrm{Spec}(T) = \mathrm{Spec}(S)\) for any subring \(T\) of \(S\) properly containing \(R\). The aim of this paper is to provide a characterization of this class of rings. Among the main results of the present work, we mention the following: (1) if \(S\) is not local, then \(R\) is a maximal non-prime ideally equal subring of \(S\) if and only if \(R \subset S\) is a minimal ring extension; (2) if \(S\) is local and \(R\) is not local, then \(R\) is a maximal non-prime ideally equal subring of \(S\) if and only if \(R \subset S\) is a closed minimal extension. The case of when both \(S\) and \(R\) are local is also characterized.