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Radical Ideals in Valuation Domains
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2007