Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T06:31:54.043Z Has data issue: false hasContentIssue false
  • English
  • Français

On PIC(D[α]) For a Principal Ideal Domain D

Published online by Cambridge University Press:  20 November 2018

Robert Gilmer  and
William Heinzer
Robert Gilmer
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027
William Heinzer
Affiliation:
Department of Mathematics, Purdue University, W. Lafayette, Indiana 47907
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let D be a PID with infinitely many maximal ideals. J. W. Brewer has asked whether some simple ring extension D[α] of D must have nontrivial Picard group. We show that this question has a negative answer.

Keywords

13B9913F10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 1 , 01 March 1989 , pp. 114 - 116
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Brewer, J., Bunce, J., and Van Vleck, F., Linear systems over commutative rings, Dekker, New York, 1986.Google Scholar
2. Brewer, J., Katz, D., and W. Ullery, Pole assignability in polynomial rings, power series rings and Prùfer domains, Algebra, J. 106 (1987), 265- 286.Google Scholar
3. Brewer, J., Klingler, L., and F. Minnaar, Polynomial rings which are BCS-rings, Comm. in Algebra (to appear).Google Scholar
4. Claborn, L., A generalized approximation theorem for Dedekind domains, Proc. Amer. Math. Soc. 18 (1967), 378–380. Google Scholar
5. Fossum, R., The divisor class group of a Krull domain, Springer-Verlag, New York, 1973.Google Scholar
6. Gilmer, R. and W. Heinzer, On the number of generators of an invertible ideal, J. Algebra 14 (1970), 139–151. Google Scholar
7. Hautus, M. and E. Sontag, New results on pole-shifting for parameterized families of systems, J. Pure Appl. Algebra 40 (1986), 229- 244.Google Scholar
8. Heitmann, R. C. and L. S. Levy, 1 1/2 and 2 generator ideals in Prufer domains, Rocky Mt. J. Math. 5 (1975), 361–373. Google Scholar
9. Lazard, D. and P. Huet, Dominions des anneaux commutatifs, Bull Sci. Math. 94 (1970), 193–199. Google Scholar
10. Nagata, M., Local rings, Interscience, New York, 1962.Google Scholar
11. Vasconcelos, W. and C. Weibel, BCS-Rings, J. Pure Appl. Algebra, to appear.Google Scholar
12. Wiegand, R. and S. Wiegand, Decomposition of torsion-free modules over affine curves, Proc. Bowdoin Conf. on Algebraic Geometry (to appear).Google Scholar