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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Catenarity in module-finite algebras
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by Shiro Goto and Kenji Nishida PDF
Proc. Amer. Math. Soc. 127 (1999), 3495-3502 Request permission

Abstract:

The main theorem says that any module-finite (but not necessarily commutative) algebra $\Lambda$ over a commutative Noetherian universally catenary ring $R$ is catenary. Hence the ring $\Lambda$ is catenary if $R$ is Cohen-Macaulay. When $R$ is local and $\Lambda$ is a Cohen-Macaulay $R$-module, we have that $\Lambda$ is a catenary ring, $\dim \Lambda =\dim \Lambda /Q+\mathrm {ht}_\Lambda Q$ for any $Q\in \operatorname {Spec}\Lambda$, and the equality $n=\mathrm {ht}_\Lambda Q- \mathrm {ht}_\Lambda P$ holds true for any pair $P\subseteq Q$ of prime ideals in $\Lambda$ and for any saturated chain $P=P_0\subset P_1\subset \cdots \subset P_n=Q$ of prime ideals between $P$ and $Q$.
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Additional Information
  • Shiro Goto
  • Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-71, Japan
  • MR Author ID: 192104
  • Email: goto@math.meiji.ac.jp
  • Kenji Nishida
  • Affiliation: Department of Mathematics, Faculty of Science, Shinsyu University, Matsumoto, 390-0802 Japan
  • Email: kenisida@math.shinsyu-u.ac.jp
  • Received by editor(s): October 27, 1997
  • Received by editor(s) in revised form: February 24, 1998
  • Published electronically: May 13, 1999
  • Additional Notes: The first author was supported by the Grant-in-Aid for Scientific Researches (C)
  • Communicated by: Ken Goodearl
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3495-3502
  • MSC (1991): Primary 13E05, 16A18; Secondary 13H10, 16A33
  • DOI: https://doi.org/10.1090/S0002-9939-99-04962-X
  • MathSciNet review: 1618674