Integral morphism: Difference between revisions
(New page: {{curing-homomorphism property}} ==Definition== Suppose <math>R</math> and <math>S</math> are commutative unital rings and <math>f:R \to S</math> is a [[homomorphism of commutative u...) |
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Suppose <math>R</math> and <math>S</math> are [[commutative unital ring]]s and <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]]. Then, we say that <math>f</math> is an ''integral morphism'' if <math>S</math> is an [[integral extension]] of the image of <math>R</math> in <math>S</math>. Equivalently, we say that <math>f</math> is an integral morphism if every element of <math>S</math> satisfies a monic polynomial with coefficients in <math>R</math>. | Suppose <math>R</math> and <math>S</math> are [[commutative unital ring]]s and <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]]. Then, we say that <math>f</math> is an ''integral morphism'' if <math>S</math> is an [[integral extension]] of the image of <math>R</math> in <math>S</math>. Equivalently, we say that <math>f</math> is an integral morphism if every element of <math>S</math> satisfies a monic polynomial with coefficients in <math>R</math>. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Finite morphism]] | |||
Revision as of 21:43, 2 February 2008
This article defines a property that can be evaluated for a homomorphism of commutative unital rings
Definition
Suppose and are commutative unital rings and is a homomorphism of commutative unital rings. Then, we say that is an integral morphism if is an integral extension of the image of in . Equivalently, we say that is an integral morphism if every element of satisfies a monic polynomial with coefficients in .