Radical Ideals in Valuation Domains

Berg, John E. van den

Canadian journal of mathematics, Tome 59 (2007) no. 4, pp. 880-896

Résumé

An ideal $I$ of a ring $R$ is called a radical ideal if $I\,=\,\mathcal{R}(R)$ where $\mathcal{R}$ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that if $R$ is a valuation domain, then a proper ideal $I$ of $R$ is a radical ideal if and only if $I$ is a distinguished ideal of $R$ (the latter property means that if $J$ and $K$ are ideals of $R$ such that $J\,\subset \,I\,\subset \,K$ then we cannot have $I/J\,\cong \,K/I$ as rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.

Détail

DOI : 10.4153/CJM-2007-037-6

Mots-clés : 16N80, 13A18

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