Projective equivalence of ideals in Noetherian integral domains

JOURNAL OF ALGEBRA

卷:320

Article

Heinzer, William J.¹  Ratliff, Louis J., Jr.²  Rush, David E.² 

[1] Purdue Univ, Dept Math, W Lafayette, IN 47907 USA

[2] Univ Calif Riverside, Dept Math, Riverside, CA 92521 USA

关键词: projective equivalence of ideals;    Rees valuations;    Rees integers;    Noetherian integral domains;   

DOI  :  10.1016/j.jalgebra.2008.06.018

来源: Elsevier

【 摘 要 】

Let I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of a finite separable integral extension domain A of R and a positive integer in such that all the Rees integers of I A are equal to in. Moreover, if R has altitude one, then all the Rees integers of J=Rad(I A) are equal to one and the ideals J(m) and I A have the same integral closure. Thus Rad(I A)=J is a projectively full radical ideal that is projectively equivalent to I A. In particular, if R is Dedekind, then there exists a Dedekind domain A having the following properties: (i) A is a finite separable integral extension of R; and (ii) there exists a radical ideal J of A and a positive integer m such that I A = J(m). In this case the extension A also has the property that for each maximal ideal N of A with I C N, the canonical inclusion R/(N boolean AND R) hooked right arrow A/N is an isomorphism, and the integer in is a multiple of [A((0)) : R-(0)]. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free