Monoidal extensions of a locally quasi-unmixed UFD

JOURNAL OF ALGEBRA

卷:431

Article

Oeser, Paul Richard¹  Rush, David E.² 

[1] Odessa Coll, Dept Math, Odessa, TX 79764 USA

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

关键词: Monoidal transform;    Unique factorization domain;    Locally quasi-unmixed Noetherian ring;    Asymptotic sequence;    Rees ring;    Integrally closed ideal;    Ideal of the principal class;   

DOI  :  10.1016/j.jalgebra.2015.01.029

来源: Elsevier

【 摘 要 】

Let R be a locally quasi-unmixed domain, a, b(1),..., b(n) an asymptotic sequence in R, I = (a,b(1),...,b(n))R and S = R[b(1)/a,..., b(n)/a]= R[I/a], the monoidal transform of R with respect to I. It is shown that S is a locally quasi-unmixed domain, a, b(1)/a,..., b(n)/a is an asymptotic sequence in S and there is a one-to-one correspondence between the asymptotic primes (A) over cap*(I) of I and the asymptotic primes (A) over cap* (aS) of aS = IS. Moreover, if a, b(1),...,b(n). is an R-sequence, this extends to a one-to-one correspondence between Ass(R)(R/I) and Asss(S/aS). In the case that R is a unique factorization domain, the height one prime ideals of S are examined to determine how far S is from being a UFD. A complete description is given of which height one prime ideals P of S are principal or have a principal primary ideal in the case that P boolean AND R has height 1. If the prime divisors of a satisfy a mild condition, a similar description is given in the case that P boolean AND R has height > 1. These are applied to give similar results for the Rees ring R[1/t, It] where t is an indeterminate. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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