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)

Request Permissions   Purchase Content 
 
 

 

A Dedekind-Mertens theorem for power series rings


Authors: Neil Epstein and Jay Shapiro
Journal: Proc. Amer. Math. Soc. 144 (2016), 917-924
MSC (2010): Primary 13F25, 13A15
DOI: https://doi.org/10.1090/proc/12661
Published electronically: November 18, 2015
MathSciNet review: 3447645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature.


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

  • [AK96] D. D. Anderson and B. G. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra 181 (1996), no. 1, 82-94. MR 1382027 (97c:13014), https://doi.org/10.1006/jabr.1996.0110
  • [And00] D. D. Anderson, GCD domains, Gauss' lemma, and contents of polynomials, Non-Noetherian commutative ring theory, Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 1-31. MR 1858155 (2002g:13039)
  • [Bre81] James W. Brewer, Power series over commutative rings, Lecture Notes in Pure and Applied Mathematics, vol. 64, Marcel Dekker, Inc., New York, 1981. MR 612477 (82i:13002)
  • [CVV98] Alberto Corso, Wolmer V. Vasconcelos, and Rafael H. Villarreal, Generic Gaussian ideals, J. Pure Appl. Algebra 125 (1998), no. 1-3, 117-127. MR 1600012 (98m:13014), https://doi.org/10.1016/S0022-4049(97)80001-1
  • [Ded92] Richard Dedekind, Über einen arithmetischen Satz von Gauß, Mittheilungen der Deutschen Mathematischen Gesellschaft in Prag, Tempsky, 1892, pp. 1-11.
  • [GGP75] Robert Gilmer, Anne Grams, and Tom Parker, Zero divisors in power series rings, J. Reine Angew. Math. 278/279 (1975), 145-164. MR 0387274 (52 #8117)
  • [GS11] Sarah Glaz and Ryan Schwarz, Prüfer conditions in commutative rings, Arab. J. Sci. Eng. 36 (2011), no. 6, 967-983 (English, with English and Arabic summaries). MR 2845525, https://doi.org/10.1007/s13369-011-0049-5
  • [HH98] William Heinzer and Craig Huneke, The Dedekind-Mertens lemma and the contents of polynomials, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1305-1309. MR 1425124 (98j:13003), https://doi.org/10.1090/S0002-9939-98-04165-3
  • [Mer92] Franz Mertens, Über einen algebraischen Satz, S.B. Akad. Wiss. Wien (2a) 101 (1892), 1560-1566.
  • [Prü32] Heinz Prüfer, Untersuchungen über Teilbarkeitseigenschaften in Körpern., J. Reine Angew. Math. 168 (1932), 1-36.
  • [Rus78] David E. Rush, Content algebras, Canad. Math. Bull. 21 (1978), no. 3, 329-334. MR 511581 (81h:13006), https://doi.org/10.4153/CMB-1978-057-8
  • [Tsa65] Hwa Tsang, Gauss's lemma, Ph.D. thesis, University of Chicago, 1965.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13F25, 13A15

Retrieve articles in all journals with MSC (2010): 13F25, 13A15


Additional Information

Neil Epstein
Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
Email: nepstei2@gmu.edu

Jay Shapiro
Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
Email: jshapiro@gmu.edu

DOI: https://doi.org/10.1090/proc/12661
Keywords: Dedekind-Mertens lemma, content of a power series
Received by editor(s): February 5, 2014
Received by editor(s) in revised form: October 8, 2014
Published electronically: November 18, 2015
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society