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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Three theorems on form rings

Author: Louis J. Ratliff
Journal: Proc. Amer. Math. Soc. 99 (1987), 432-436
MSC: Primary 13E05; Secondary 13A17, 13C15
MathSciNet review: 875376
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Abstract: Three theorems concerning the form ring (= associated graded ring) $ {\mathbf{F}}\left( {R,I} \right)$ of an ideal $ I$ in a Noetherian ring $ R$ are proved. The first characterizes, for a $ P$-primary ideal in a locally quasi-unmixed ring, when $ {\mathbf{F}}{\left( {R,I} \right)_{\operatorname{red}}}$ is an integral domain in terms of when $ {\mathbf{F}}{\left( {{R_P},I{R_P}} \right)_{\operatorname{red}}}$ is an integral domain. For an aribtrary Noetherian ring $ R$ the second gives a somewhat similar characterization for $ {\mathbf{F}}\left( {R,J} \right)$ to have only one prime divisor of zero for some ideal $ J$ that is projectively equivalent to $ I$. And the third characterizes unmixed semilocal rings in terms of the existence of an open ideal $ I$ such that the zero ideal in $ {\mathbf{F}}\left( {R,I} \right)$ is isobathy.

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Keywords: Analytically unramified semilocal ring, asymptotic prime divisor, analytic spread of an ideal, essential prime divisor, form ring, integral closure of an ideal, isobathy ideal, Noetherian ring, normal ideal, projectively equivalent ideals, quasi-unmixed local ring, Rees ring, unmixed local ring, $ u$-essential prime divisor
Article copyright: © Copyright 1987 American Mathematical Society

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