Idempotent Pairs and PRINC Domains

A pair of elements a, b in an integral domain R is an idempotent pair if either \(a(1-a) \in bR\), or \(b(1-b) \in aR\). R is said to be a PRINC domain if all the ideals generated by an idempotent pair are principal. We show that in an order R of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if R is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. Furthermore, we show that the only imaginary quadratic orders \(\mathbb {Z}[\sqrt{-d}]\), \(d > 0\) square-free, that are PRINC and not integrally closed, are for \(d=3,7\).