Radically closed subring - Revision history

Vipul at 19:02, 6 February 2009

2009-02-06T19:02:25Z

← Older revision		Revision as of 19:02, 6 February 2009
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	* [[Weaker than::Integrally closed subring]]		* [[Weaker than::Integrally closed subring]]
	* [[Weaker than::Algebraically 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.

Vipul at 19:00, 6 February 2009

2009-02-06T19:00:06Z

← Older revision		Revision as of 19:00, 6 February 2009
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	{{wikilocal}}		{{wikilocal}}
	{curing-subring property}}		{{curing-subring property}}

	==Definition==		==Definition==

Vipul: New page: {{wikilocal}} {curing-subring property}} ==Definition== A unital subring $S$ of a commutative unital ring $R$ is termed '''radically closed''' if for every ...

2009-02-06T18:59:49Z

New page: {{wikilocal}} {curing-subring property}} ==Definition== A unital subring <math>S</math> of a commutative unital ring <math>R</math> is termed '''radically closed''' if for every ...

New page

{{wikilocal}}
{curing-subring property}}

==Definition==

A [[unital subring]] <math>S</math> of a [[commutative unital ring]] <math>R</math> is termed '''radically closed''' if for every <math>x \in R</math> such that there exists <math>n \ge 1</math> for which <math>x^n \in S</math>, we have <math>x \in S</math>.

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Integrally closed subring]]
* [[Weaker than::Algebraically closed subring]]

Radically closed subring - Revision history

Vipul at 19:02, 6 February 2009

Vipul at 19:00, 6 February 2009

Vipul: New page: {{wikilocal}} {curing-subring property}} ==Definition== A unital subring S of a commutative unital ring R is termed '''radically closed''' if for every ...

Vipul: New page: {{wikilocal}} {curing-subring property}} ==Definition== A unital subring $S$ of a commutative unital ring $R$ is termed '''radically closed''' if for every ...