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

DOI:
https://doi.org/10.1090/S0002-9939-01-06231-1

Published electronically:
October 24, 2001

MathSciNet review:
1887039

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

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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

DOI:
https://doi.org/10.1090/S0002-9939-01-06231-1

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