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Two notes on ideal-transforms

Published online by Cambridge University Press:  24 October 2008

Daniel Katz  and
Louis J. Ratliff Jr
Daniel Katz
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A.
Louis J. Ratliff Jr
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, U.S.A.
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Abstract

The first note gives two new characterization of the ideal-transform T(I) of a finitely generated regular ideal I in a large class of rings. Specifically, if b is a regular element in I, then there exists a regular element c ∈ I and a multiplicatively closed set S of regular elements in R such that T(I) = T((b, c)R) = Rb ∩ Rc = Rb ∩ Rs, so T(I) is the ideal-transform of an ideal generated by two elements, and every ring of the form Rb ∩ Rs is an ideal-transform. The second theorem shows that if T(I) is integrally closed, then it is a Krull ring. As an application of these results we strengthen some known results concerning when certain ideal-transforms of the Rees ring R(R, I) are finite or integral extension rings of R(R, I).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 389 - 397
Copyright
Copyright © Cambridge Philosophical Society 1987

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