The homogeneous spectrum of a graded commutative ring

Heinzer, William; Roitman, Moshe

doi:10.1090/S0002-9939-01-06231-1

The homogeneous spectrum of a graded commutative ring
HTML articles powered by AMS MathViewer

by William Heinzer and Moshe Roitman PDF

Proc. Amer. Math. Soc. 130 (2002), 1573-1580 Request permission

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 $\operatorname {rank} \Gamma \le 2$, 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

David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
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
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
S. Mori, Über eindeutige Reduktion von Idealen in Ringen ohne Teilerkettensatz, J. Sci. Hiroshima Univ. Ser. A 3(1933), 275-318.
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
Jack Ohm and R. L. Pendleton, Rings with noetherian spectrum, Duke Math. J. 35 (1968), 631–639. MR 229627
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335