Idempotent Pairs and PRINC domains

Abstract

A pair of elements a, b in an integral domain R is an idempotent pair if either a(1 − a) ∈ bR, or b(1 − b) ∈ 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 ℤ[√−d], d > 0 square-free, that are PRINC and not integrally closed, are for d = 3, 7.