Idempotent pairs and PRINC domains

TL;DR

This paper investigates the properties of PRINC domains, showing that in Dedekind domains, being PRINC is equivalent to being a PID, and characterizes certain quadratic orders that are PRINC but not integrally closed.

Contribution

It establishes a characterization of PRINC domains within Dedekind domains and identifies specific quadratic orders that are PRINC but not integrally closed.

Findings

Dedekind domains are PRINC if and only if they are PIDs.

Every regular prime ideal in an order of a Dedekind domain can be generated by an idempotent pair.

Only the quadratic orders for d=3,7 are PRINC but not integrally closed.

Abstract

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 $d=3,7$.