Characterizing when is integrally closed

Author:
Thomas G. Lucas

Journal:
Proc. Amer. Math. Soc. **105** (1989), 861-867

MSC:
Primary 13B20; Secondary 13B30

MathSciNet review:
942636

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Abstract: Unlike the situation when dealing with integral domains, it is not always the case that the polynomial ring is integrally closed when is an integrally closed commutative ring with nonzero zero divisors. In the main theorem it is shown that for an integrally closed reduced ring , is not integrally closed if and only if there exists a finitely generated dense ideal and an -module homomorphism such that is integral over and is not defined by multiplication by a fixed element of . As a corollary it is shown that is integrally closed if and only if is integrally closed in , the total quotient ring of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0942636-0

Keywords:
Reduced ring,
integrally closed,
complete ring of quotients,
dense ideal

Article copyright:
© Copyright 1989
American Mathematical Society