Let be a ring and a subring of . We say that an element is integral over if there is a monic polynomial , with each so that . The elements of that are integral over form a subring of which contains . We call the integral closure of in . If , then we say that is integral over . If , then we say that is integrally closed in . If is integrally closed in its field of fractions, then we call it integrally closed.
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