Radically closed subring: Difference between revisions

Latest revision as of 19:02, 6 February 2009

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This article defines a property that can be evaluated for a unital subring in a commutative unital ring: given any commutative unital ring and a subring thereof, the property is either true or false for the pair
View a complete list of such properties

Definition

A unital subring $S$ of a commutative unital ring $R$ is termed radically closed if for every $x \in R$ such that there exists $n \geq 1$ for which $x^{n} \in S$ , we have $x \in S$ .

Relation with other properties

Stronger properties

Related properties

A radical ideal is an ideal with the analogous property: if $x^{n}$ is in the ideal, so is $x$ . The radical of an ideal is the smallest ideal containing it that is a radical ideal. It turns out that for any ideal, every element in its radical has the property that some power of it is in the ring. For full proof, refer: Equivalence of definitions of radical of an ideal

Thus, if an ideal is contained in a radically closed subring, the radical of that ideal is also contained in that subring.

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 * [[Weaker than::Integrally closed subring]]
 * [[Weaker than::Algebraically closed subring]]
+===Related properties===
+A [[radical ideal]] is an ideal with the analogous property: if <math>x^n</math> is in the ideal, so is <math>x</math>. The [[radical of an ideal]] is the smallest ideal containing it that is a [[radical ideal]]. It turns out that for any ideal, every element in its radical has the property that some power of it is in the ring. {{proofat|[[Equivalence of definitions of radical of an ideal]]}}
+Thus, if an ideal is contained in a radically closed subring, the radical of that ideal is also contained in that subring.