The homogeneous spectrum of a graded commutative ring
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- 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
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Additional Information
- William Heinzer
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
- Email: heinzer@math.purdue.edu
- Moshe Roitman
- Affiliation: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
- Email: mroitman@math.haifa.ac.il
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1573-1580
- MSC (1991): Primary 13A15, 13E99
- DOI: https://doi.org/10.1090/S0002-9939-01-06231-1
- MathSciNet review: 1887039