Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Coprime packedness and set theoretic complete intersections of ideals in polynomial rings


Author: V. Erdogdu
Journal: Proc. Amer. Math. Soc. 132 (2004), 3467-3471
MSC (2000): Primary 13B25, 13B30, 13C15, 13C20; Secondary 13A15, 13A18
DOI: https://doi.org/10.1090/S0002-9939-04-07438-6
Published electronically: July 14, 2004
MathSciNet review: 2084066
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A ring $R$ is said to be coprimely packed if whenever $I$ is an ideal of $R$and $S$ is a set of maximal ideals of $R$ with $I\subseteq\bigcup\{M\in S\}$, then $I\subseteq M$ for some $M\in S$. Let $R$ be a ring and $R\langle X\rangle$ be the localization of $R[X]$ at its set of monic polynomials. We prove that if $R$ is a Noetherian normal domain, then the ring $R\langle X\rangle$ is coprimely packed if and only if $R$ is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of $R[X]$ is a set theoretic complete intersection. A similar result is also proved when $R$ is either a Noetherian arithmetical ring or a Bézout domain of dimension one.


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

  • 1. J. W. Brewer and W. J. Heinzer, $R$ Noetherian implies $R\langle X\rangle$ is a Hilbert ring, J. Algebra, 67 (1980), 204-209. MR 82d:13010
  • 2. V. Erdogdu, Coprimely packed rings, J. Number Theory, 28 (1988), 1-5. MR 89f:13025
  • 3. V. Erdogdu, The prime avoidance of ideals in Noetherian Hilbert rings, Communications in Algebra, 22 (1994), 4989-4990.
  • 4. V. Erdogdu, Three notes on coprime packedness, J. Pure Appl. Algebra, 148 (2000), 165-170. MR 2001b:13004
  • 5. V. Erdogdu and S. McAdam, Coprimely packed Noetherian polynomial rings, Communications in Algebra, 22 (1994), 6459-6470. MR 95k:13025
  • 6. S. Glaz and W. Vasconcelos, The content of Gaussian polynomials, J. Algebra, 202 (1998), 1-9. MR 99c:13003
  • 7. I. Kaplansky, Commutative Rings, University of Chicago Press, 1974. MR 49:10674
  • 8. L. Le Riche, The ring $R\langle X\rangle$, J. Algebra, 67 (1980), 327-341. MR 82d:13011
  • 9. D. E. Rush, Generating ideals up to radical in Noetherian polynomial rings, Communications in Algebra, 25 (1997), 2169-2191. MR 98i:13041

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13B25, 13B30, 13C15, 13C20, 13A15, 13A18

Retrieve articles in all journals with MSC (2000): 13B25, 13B30, 13C15, 13C20, 13A15, 13A18


Additional Information

V. Erdogdu
Affiliation: Department of Mathematics, Istanbul Technical University, Maslak, 80626 Istanbul, Turkey
Email: erdogdu@itu.edu.tr

DOI: https://doi.org/10.1090/S0002-9939-04-07438-6
Keywords: Coprime packedness, polynomial rings, class group, set theoretic complete intersection
Received by editor(s): July 17, 2002
Received by editor(s) in revised form: June 25, 2003
Published electronically: July 14, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society