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arXiv · 1412.8089

Idempotent pairs and PRINC domains

Giulio Peruginelli·Luigi Salce·Paolo Zanardo
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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 $d=3,7$.

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Giulio Peruginelli, Luigi Salce, Paolo Zanardo. 2014-12-27. Idempotent pairs and PRINC domains. https://doi.org/10.1007/978-3-319-38855-7_13

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