Hostname: page-component-7dd5485656-frp75 Total loading time: 0 Render date: 2025-10-21T21:15:00.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Elasticity in Polynomial-Type Extensions

Part of: Arithmetic rings and other special rings

Published online by Cambridge University Press:  28 March 2016

Mark Batell  and
Jim Coykendall
Mark Batell
Affiliation:
Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA
Jim Coykendall
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA (jcoyken@clemson.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domain R and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every non-constant irreducible polynomial f ∈ R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.

Keywords

factorizationunique factorizationpolynomialsAP-domainGL-domain

MSC classification

Secondary: 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half-factorial)

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 581 - 590
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Anderson, D. D. and Quintero, R. O., Some generalizations of GCD-domains, in Factorization in integral domains, Lecture Notes in Pure and Applied Mathematics, Volume 189, pp. 189–195 (Dekker, New York, 1997).Google Scholar
2. Arnold, J. and Sheldon, P., Integral domains that satisfy Gauss's lemma, Michigan Math. J. 22 (1975), 39–51.Google Scholar
3. Carlitz, L., A characterization of algebraic number fields with class number two, Proc. Am. Math. Soc. 11 (1960), 391–392.Google Scholar
4. Chapman, S. T. and Smith, W. W., Restricted elasticity and rings of integer-valued polynomials determined by finite subsets, Monatsh. Math. 148(3) (2006), 195–203.Google Scholar
5. Cohn, H., Advanced number theory (Dover, New York, 1980).Google Scholar
6. Coykendall, J., Half-factorial domains in quadratic fields, J. Alg. 235 (2001), 417–430.Google Scholar
7. Coykendall, J., A characterization of polynomial rings with the half-factorial property, in Factorization in integral domains, Lecture Notes in Pure and Applied Mathematics, Volume 189, pp. 291–294 (Dekker, New York, 1997).Google Scholar
8. Coykendall, J. and Mammenga, B., An embedding theorem, J. Alg. 325 (2011), 177–185.Google Scholar
9. Coykendall, J. and Zafrullah, M., AP-domains and unique factorization, J. Pure Appl. Alg. 189 (2004), 27–35.Google Scholar
10. Gilmer, R., Multiplicative ideal theory (Dekker, New York, 1972).Google Scholar
11. Tang, H., Gauss’ lemma, Proc. Am. Math. Soc. 35 (1972), 372–376.Google Scholar
12. Zaks, A., Half-factorial domains, Israel J. Math. 37 (1980), 281–302.Google Scholar