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In algebra, the integral closure of an ideal I of a commutative ring R, denoted by I ¯ {\displaystyle {\overline {I}}} , is the set of all elements r in R that are integral over I: there exist a i ∈ I i {\displaystyle a_{i}\in I^{i}} 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 I ¯ {\displaystyle {\overline {I}}} if and only if there is a finitely generated R-module M, annihilated only by zero, such that r M ⊂ I M {\displaystyle rM\subset IM}. It follows that I ¯ {\displaystyle {\overline {I}}} is an ideal of R I is said to be integrally closed if I = I ¯ {\displaystyle I={\overline {I}}}.
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.