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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Monoidal extensions of a Cohen-Macaulay unique factorization domain
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by William J. Heinzer, Aihua Li, Louis J. Ratliff Jr. and David E. Rush PDF
Trans. Amer. Math. Soc. 354 (2002), 1811-1835 Request permission

Abstract:

Let $A$ be a Noetherian Cohen-Macaulay domain, $b$, $c_1$, $\dots$, $c_g$ an $A$-sequence, $J$ = $(b,c_1,\dots ,c_g)A$, and $B$ = $A[J/b]$. Then $B$ is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets $\operatorname {Ass}_B(B/bB)$ and $\operatorname {Ass}_A(A/J)$, and each $q$ $\in$ $\operatorname {Ass}_A(A/J)$ has height $g+1$. If $B$ does not have unique factorization, then some height-one prime ideals $P$ of $B$ are not principal. These primes are identified in terms of $J$ and $P \cap A$, and we consider the question of how far from principal they can be. If $A$ is integrally closed, necessary and sufficient conditions are given for $B$ to be integrally closed, and sufficient conditions are given for $B$ to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if $P$ is a height-one prime ideal of $B$, then $P \cap A$ also has height one if and only if $b$ $\notin$ $P$ and thus $P \cap A$ has height one for all but finitely many of the height-one primes $P$ of $B$. If $A$ has unique factorization, a description is given of whether or not such a prime $P$ is a principal prime ideal, or has a principal primary ideal, in terms of properties of $P \cap A$. A similar description is also given for the height-one prime ideals $P$ of $B$ with $P \cap A$ of height greater than one, if the prime factors of $b$ satisfy a mild condition. If $A$ is a UFD and $b$ is a power of a prime element, then $B$ is a Krull domain with torsion class group if and only if $J$ is primary and integrally closed, and if this holds, then $B$ has finite cyclic class group. Also, if $J$ is not primary, then for each height-one prime ideal $p$ contained in at least one, but not all, prime divisors of $J$, it holds that the height-one prime $pA[1/b] \cap B$ has no principal primary ideals. This applies in particular to the Rees ring ${\mathbf R}$ $=$ $A[1/t, tJ]$. As an application of these results, it is shown how to construct for any finitely generated abelian group $G$, a monoidal transform $B$ = $A[J/b]$ such that $A$ is a UFD, $B$ is Cohen-Macaulay and integrally closed, and $G$ $\cong$ $\operatorname {Cl}(B)$, the divisor class group of $B$.
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Additional Information
  • William J. Heinzer
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47909-1395
  • Email: heinzer@math.purdue.edu
  • Aihua Li
  • Affiliation: Department of Mathematics and Computer Science, Loyola University, New Orleans, Louisiana 70118
  • Email: ali@loyno.edu
  • Louis J. Ratliff Jr.
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
  • Email: ratliff@newmath.ucr.edu
  • David E. Rush
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
  • Email: rush@newmath.ucr.edu
  • Received by editor(s): December 31, 2000
  • Published electronically: January 9, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 1811-1835
  • MSC (2000): Primary 13A05, 13A30, 13B02, 13B22, 13C20, 13F15, 13H10
  • DOI: https://doi.org/10.1090/S0002-9947-02-02951-3
  • MathSciNet review: 1881018