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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.

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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