About: Integral closure of an ideal

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In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to if and only if there is a finitely generated R-module M, annihilated only by zero, such that . It follows that is an ideal of R (in fact, the integral closure of an ideal is always an ideal; see below.) I is said to be integrally closed if .

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  • In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to if and only if there is a finitely generated R-module M, annihilated only by zero, such that . It follows that is an ideal of R (in fact, the integral closure of an ideal is always an ideal; see below.) I is said to be integrally closed if . The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring. (en)
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  • In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to if and only if there is a finitely generated R-module M, annihilated only by zero, such that . It follows that is an ideal of R (in fact, the integral closure of an ideal is always an ideal; see below.) I is said to be integrally closed if . (en)
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  • Integral closure of an ideal (en)
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