Idempotent pairs and PRINC domains

Abstract

A pair of elements $a,b$ in an integral domain $R$ is an idempotent pair if either $a(1-a)\in bR$, 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 $d=3,7$.

Updated references

Cited by in corpus (3)

• Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings
• PRINC domains and comaximal factorization domains
• The lattice of primary ideals of orders in quadratic number fields