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On the Graph of Divisibility of an Integral Domain

Published online by Cambridge University Press:  20 November 2018

Jason Greene Boynton  and
Jim Coykendall
Jason Greene Boynton
Affiliation:
North Dakota State University, Department of Mathematics, Fargo, ND, USA e-mail: [email protected]
Jim Coykendall
Affiliation:
Mathematical Sciences, Clemson University, Martin O-103, Clemson, SC 29634-0975, USA e-mail: [email protected]
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Abstract

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It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

Keywords

13F1513A05atomicfactorizationdivisibility
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 3 , 01 September 2015 , pp. 449 - 458
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Anderson, D. F., Anderson, D. F., and Zafrullah, M. Factorization in integral domains. J. Pure Appl. Algebra. 69(1990), no. 1, 1–19. http://dx.doi.org/10.1016/0022-4049(90)90074-R Google Scholar
[2] Arenas, F. G., Alexandroff spaces. Acta Math.Univ. Comenian. 68(1999), no. 1,17–25.Google Scholar
[3] Chapman, S. T. and Smith, W. W., Restricted elasticity in rings of integer-valued polynomials determined by finite subsets. Monatsh.Math.148(2006), no. 3,195–203. http://dx.doi.org/10.1007/s00605-005-0357-5 Google Scholar
[4] Cohn, P. M., Bézout rings and their subrings. Proc. Cambridge Philos. Soc. 64(1968), 251–264. http://dx.doi.Org/1 0.101 7/S0305004100042791 Google Scholar
[5] Grams, A., Atomic domains and the ascending chain condition. Proc. Cambridge Philos. Soc. 75(1974), 321–329. http://dx.doi.org/10.1017/S0305004100048532 Google Scholar
[6] Mott, J. L., The group of divisibility and its applications. In: Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., 311, Springer, Berlin, 1973, pp.199–224.Google Scholar
[7] Zaks, A., Atomic rings without a.c.c. on principal ideals. J. Algebra 74 (1982), no. 1, 223–231. http://dx.doi.org/!0.1016/0021-8693(82)90015-1 Google Scholar