Journal of Commutative Algebra

Normsets of almost Dedekind domains and atomicity

Richard Erwin Hasenauer

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we will introduce a new norm map on almost Dedekind domains. We compare and contrast our new norm map to the traditional Dedekind-Hasse norm. We prove that factoring in an almost Dedekind domain is in one-to-one correspondence to factoring in the new normset, improving upon this results in \cite {Coykendall}. In \cite {Grams}, an atomic almost Dedekind domain was constructed with a trivial Jacobson radical. We pursue atomicity in almost Dedekind domains with nonzero Jacobson radicals, showing the usefulness of the new norm we introduced. We state theorems with regard to specific classes of almost Dedekind domains. We provide a necessary condition for an almost Dedekind domain with nonzero Jacobson radical to be atomic.

Article information

J. Commut. Algebra, Volume 8, Number 1 (2016), 61-75.

First available in Project Euclid: 28 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]
Secondary: 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05]

Factorization almost Dedekind


Hasenauer, Richard Erwin. Normsets of almost Dedekind domains and atomicity. J. Commut. Algebra 8 (2016), no. 1, 61--75. doi:10.1216/JCA-2016-8-1-61.

Export citation


  • J. Coykendall, Normsets and determination of unique factorization in rings of algebraic integers, Proc. Amer. Math. Soc. 124 (1996), 1727–1732.
  • J. Coykendall, D. Dobbs and B. Mullins, On integral domains with no atoms, Comm. Alg. 27 (1999), 5813–5831.
  • R. Gilmer, Multiplicative Ideal Theory, Queen's Papers Pure Appl. Math. 90, Queen's University Press, Kingston, 1992.
  • A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Camb. Phil. Soc. 75 (1974), 321–329.
  • K.A. Loper, Almost Dedekind domains which are not Dedekind, in Multiplicative ideal theory in commutative algebra; A tribute to the work of Robert Gilmer, James W. Brewer, Sarah Glaz, William J. Heinzer and Bruce M. Olberding, eds., Springer, New York, 2006.
  • ––––, Sequence domains and integer-valued polynomials, J. Pure Appl. Alg. 119 (1997), 185–210.
  • K.A. Loper and T.G. Lucas, Factoring ideals in almost Dedekind domains, J. reine angew. Math. 565 (2003), 61–78.
  • B. Olberding, Factorization into radical ideals, in Arithmetical properties of commutative rings and monoids, Lect. Notes Pure Appl. Math. 241, Chapman & Hall/CRC, Boca Raton, FL, 2005.