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On the Prime Ideals in a Commutative Ring

  • David E. Dobbs (a1)
Abstract

If n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

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References
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[1] Bourbaki, N., Algèbre Commutative. Masson, Paris, 1983, Chapîtres 8–9.
[2] Dobbs, D. E., On INC-extensions and polynomials with unit content. Canad. Math. Bull. 23 (1980), 3742.
[3] Dobbs, D. E., Integral extensions with fibers of prescribed cardinality. In: Zero-dimensional commutative rings (eds. D. F. Anderson and D. E. Dobbs), Lecture Notes in Pure and Appl. Math 171(1995), Dekker, New York, 201207.
[4] Dobbs, D. E., A going-up theorem for arbitrary chains of prime ideals. Comm. Algebra, to appear.
[5] Dobbs, D. E., Fontana, M. and Kabbaj, S., Direct limits of Jaffard domains. Comment. Math. Univ. St. Pauli. 39 (1990), 143155.
[6] Gilmer, R., Nashier, B. and Nichols, W., On the heights of prime ideals under integral extensions. Arch. Math. 52 (1989), 4752.
[7] Grothendieck, A. and Dieudonné, J. A., Éléments de Géométrie Algébrique, I. Springer-Verlag, Berlin, 1971.
[8] Halmos, P. R., Naive Set Theory. Van Nostrand, Princeton, 1960.
[9] Kang, B. Y. and Oh, D. Y., Lifting up an infinite chain of prime ideals to a valuation ring. Proc. Amer. Math. Soc. 126 (1998), 645646.
[10] Kaplansky, I., Commutative Rings. Rev. ed., University of Chicago Press, 1974.
[11] Kelley, J. L., General Topology. Van Nostrand, Princeton, 1955.
[12] Zariski, O. and Samuel, P., Commutative Algebra, I. Van Nostrand, Princeton, 1958.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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