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A note on universally zero-divisor rings

  • S. Visweswaran (a1)
Abstract

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

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References
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[1]Atiyah, M.F. and Macdonald, I.G., Introduction to commutative algebra (Addison Wesley, Reading, MA, 1969).
[2]Bourbaki, N., Commutative algebra (Addision Wesley, Reading, MA, 1972).
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[5]Gilmer, R., Multiplicative ideal theory 12, Queen's papers on Pure and Applied Mathematics (Queen's University, Kingston, Ontario, 1968).
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[8]Gilmer, R. and Huckaba, J.A., ‘The transform formula for ideals’, J. Algebra 21 (1972), 191215.
[9]Heinzer, W. and Lantz, D., ‘The Laskerian property in commutative rings’, J. Algebra 72 (1981), 101114.
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[13]Visweswaran, S., ‘Subrings of K[y 1,…, y t] of the type D + I’, J. Algebra 117 (1988), 374389.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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