A Dedekind-Mertens theorem for power series rings
HTML articles powered by AMS MathViewer
- by Neil Epstein and Jay Shapiro PDF
- Proc. Amer. Math. Soc. 144 (2016), 917-924 Request permission
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
- 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, DOI 10.1006/jabr.1996.0110
- 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
- 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
- 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, DOI 10.1016/S0022-4049(97)80001-1
- Richard Dedekind, Über einen arithmetischen Satz von Gauß, Mittheilungen der Deutschen Mathematischen Gesellschaft in Prag, Tempsky, 1892, pp. 1–11.
- Robert Gilmer, Anne Grams, and Tom Parker, Zero divisors in power series rings, J. Reine Angew. Math. 278(279) (1975), 145–164. MR 387274
- 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, DOI 10.1007/s13369-011-0049-5
- 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, DOI 10.1090/S0002-9939-98-04165-3
- Franz Mertens, Über einen algebraischen Satz, S.B. Akad. Wiss. Wien (2a) 101 (1892), 1560–1566.
- Heinz Prüfer, Untersuchungen über Teilbarkeitseigenschaften in Körpern., J. Reine Angew. Math. 168 (1932), 1–36.
- David E. Rush, Content algebras, Canad. Math. Bull. 21 (1978), no. 3, 329–334. MR 511581, DOI 10.4153/CMB-1978-057-8
- Hwa Tsang, Gauss’s lemma, Ph.D. thesis, University of Chicago, 1965.
Additional Information
- Neil Epstein
- Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
- MR Author ID: 768826
- Email: nepstei2@gmu.edu
- Jay Shapiro
- Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
- Email: jshapiro@gmu.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 917-924
- MSC (2010): Primary 13F25, 13A15
- DOI: https://doi.org/10.1090/proc/12661
- MathSciNet review: 3447645