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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Zero divisors in Noetherian-like rings
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by E. Graham Evans PDF
Trans. Amer. Math. Soc. 155 (1971), 505-512 Request permission

Abstract:

The zero divisors of $R/I$ for every ideal $I$ of a Noetherian ring is a finite union of primes. We take this property as a definition and study the class of rings so defined. Such rings are stable under localization and quotients. They are not stable under integral closure and are highly unstable under polynomial adjunction. The length of maximal $R$ sequences is well defined on them. In this paper all rings are commutative with unit and all modules are unitary.
References
  • N. Bourbaki, ÉlĂ©ments de mathĂ©matique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtrations et topologies. Chapitre 4: IdĂ©aux premiers associĂ©s et dĂ©composition primaire, ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1293, Hermann, Paris, 1961 (French). MR 0171800
  • Irving Kaplansky, Commutative rings, Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
  • Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
  • Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 155 (1971), 505-512
  • MSC: Primary 13.50
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0272773-9
  • MathSciNet review: 0272773