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Integral closure of an ideal

In algebra, the integral closure of an ideal I {\displaystyle I} of a commutative ring R {\displaystyle R} , denoted by I ¯ {\displaystyle {\overline {I}}} , is the set of all elements r in R {\displaystyle R} that are integral over I {\displaystyle I} : that is, for each i {\displaystyle i} there exists a i ∈ I i {\displaystyle a_{i}\in I^{i}} such that

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