Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 

 

Zero divisors in Noetherian-like rings


Author: E. Graham Evans
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtra- tions et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles, No. 1293, Hermann, Paris, 1961 (French). MR 0171800
  • [2] Irving Kaplansky, Commutative rings, Revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
  • [3] Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
  • [4] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13.50

Retrieve articles in all journals with MSC: 13.50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0272773-9
Keywords: Noetherian rings, zero divisors, integral closure, polynomials, $ R$ sequences, Serre subcategory, Laskerian, primary ideals, valuation rings
Article copyright: © Copyright 1971 American Mathematical Society