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The homogeneous spectrum of a graded commutative ring

Authors: William Heinzer and Moshe Roitman
Journal: Proc. Amer. Math. Soc. 130 (2002), 1573-1580
MSC (1991): Primary 13A15, 13E99
Published electronically: October 24, 2001
MathSciNet review: 1887039
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Abstract: Suppose $\Gamma$ is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a $\Gamma $-graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose $A$ is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring $A[\Gamma]$ have Noetherian spectrum. If $\rank \Gamma \le 2$, we show that $A[\Gamma]$ has Noetherian spectrum, while for each $n \ge 3$ we establish existence of an example where the homogeneous spectrum of $A[\Gamma]$ is not Noetherian.

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

  • 1. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
  • 2. Robert Gilmer and William Heinzer, On Jónsson modules over a commutative ring, Acta Sci. Math. (Szeged) 46 (1983), no. 1-4, 3–15. MR 739017
  • 3. Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
  • 4. Ernst Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the German by Michael Ackerman; With a preface by David Mumford. MR 789602
  • 5. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • 6. S. Mori, Über eindeutige Reduktion von Idealen in Ringen ohne Teilerkettensatz, J. Sci. Hiroshima Univ. Ser. A 3(1933), 275-318.
  • 7. 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
  • 8. Jack Ohm and R. L. Pendleton, Rings with noetherian spectrum, Duke Math. J. 35 (1968), 631–639. MR 229627
  • 9. A. Seidenberg A note on the dimension theory of rings, Pac. J. Math., 3 (1953), 505-512. MR 14:941c
  • 10. O. Zariski and P. Samuel, Commutative Algebra, volume I, Van Nostrand, New York, 1958. MR 19:833e

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Additional Information

William Heinzer
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395

Moshe Roitman
Affiliation: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Keywords: Graded ring, homogeneous spectrum, Noetherian spectrum, torsion-free cancellative commutative monoid
Received by editor(s): September 20, 2000
Received by editor(s) in revised form: December 13, 2000
Published electronically: October 24, 2001
Additional Notes: This work was prepared while the second author enjoyed the hospitality of Purdue University.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2001 American Mathematical Society