Radically closed subring: Difference between revisions
(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 ...) |
No edit summary |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{wikilocal}} | {{wikilocal}} | ||
{curing-subring property}} | {{curing-subring property}} | ||
==Definition== | ==Definition== | ||
| Line 12: | Line 12: | ||
* [[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. | |||
Latest revision as of 19:02, 6 February 2009
BEWARE! This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.
Learn more about terminology local to the wiki OR view a complete list of such terminology
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 of a commutative unital ring is termed radically closed if for every such that there exists for which , we have .
Relation with other properties
Stronger properties
Related properties
A radical ideal is an ideal with the analogous property: if is in the ideal, so is . 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.