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)

 
 

 

Finite factorization domains


Authors: D. D. Anderson and Bernadette Mullins
Journal: Proc. Amer. Math. Soc. 124 (1996), 389-396
MSC (1991): Primary 13A05, 13A15, 13E05, 13G05
DOI: https://doi.org/10.1090/S0002-9939-96-03284-4
MathSciNet review: 1322910
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An integral domain $R$ is a finite factorization domain if each nonzero element of $R$ has only finitely many divisors, up to associates. We show that a Noetherian domain $R$ is an FFD $\Leftrightarrow $ for each overring $R'$ of $R$ that is a finitely generated $R$-module, $U(R')/U(R)$ is finite. For $R$ local this is also equivalent to each $R/[R:R']$ being finite. We show that a one-dimensional local domain $(R,M)$ is an FFD $\Leftrightarrow $ either $R/M$ is finite or $R$ is a DVR.


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

  • 1 D.D. Anderson, Integral domains with finitely generated groups of divisibility, Proc. Amer. Math. Soc. 112 (1991), 613--618, MR 92c:13001.
  • 2 D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1--19, MR 92b:13028.
  • 3 ------, Factorization in integral domains II, J. Algebra 152 (1992), 78--93, MR 94c:13019.
  • 4 D.D. Anderson and J.L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements, J. Algebra 148 (1992), 17--41, MR 93e:13041.
  • 5 A. Brandis, Über die multiplikative Structure von Körpererweiterungen, Math. Z. 87 (1965), 71--73, MR 30:1124.
  • 6 R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972, MR 55:323.
  • 7 B. Glastad and J.L. Mott, Finitely generated groups of divisibility, Contemp. Math. 8 (1982), 231--247, MR 83h:13001.
  • 8 A. Grams and H. Warner, Irreducible divisors in domains of finite character, Duke Math. J. 42 (1975), 271--284, MR 51:12836.
  • 9 F. Halter-Koch, Finiteness theorems for factorizations, Semigroup Forum 44 (1992), 112--117, MR 92k:20121.
  • 10 W. Heinzer, C. Rotthaus, and J. Sally, Formal fibers and birational extensions, Nagoya Math. J. 131 (1993), 1--38, MR 95a:13008.
  • 11 I. Kaplansky, Commutative rings, University of Chicago Press, Chicago, 1974, MR 49:10674.
  • 12 K.B. Levitz and J.L. Mott, Rings with finite norms, Canad. J. Math. 24 (1972), 557--565, MR 45:6872.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13A05, 13A15, 13E05, 13G05

Retrieve articles in all journals with MSC (1991): 13A05, 13A15, 13E05, 13G05


Additional Information

D. D. Anderson
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: dan-anderson@uiowa.edu

Bernadette Mullins
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: bmullins@math.ysu.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03284-4
Keywords: Finite factorization domain (FFD)
Received by editor(s): September 1, 1994
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society