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Radical Ideals in Valuation Domains
Published online by Cambridge University Press: 20 November 2018
Abstract
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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
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