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Radical Ideals in Valuation Domains

  Published:2007-08-01
 Printed: Aug 2007
  • John E. van den
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Abstract

An ideal $I$ of a ring $R$ is called a radical ideal if $I={\mathcalR}(R)$ 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.
MSC Classifications: 16N80, 13A18 show english descriptions General radicals and rings {For radicals in module categories, see 16S90}
Valuations and their generalizations [See also 12J20]
16N80 - General radicals and rings {For radicals in module categories, see 16S90}
13A18 - Valuations and their generalizations [See also 12J20]
 

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