On irreducible divisor graphs in commutative rings with zero-divisors

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Published: Dec 22, 2015
DOI: https://doi.org/10.5556/j.tkjm.46.2015.1753
Keywords:
factorization zero-divisors commutative rings zero-divisor graphs irreducible divisor graphs

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Christopher Park Mooney
Westminster College

Abstract

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in integral domains. This inspired the so called irreducible divisor graph of an integral domain studied by J. Coykendall and J. Maney. Factorization in rings with zero-divisors is considerably more complicated than integral domains and has been widely studied recently. We find that many of the same techniques can be extended to rings with zero-divisors. In this article, we construct several distinct irreducible divisor graphs of a commutative ring with zero-divisors. This allows us to use graph theoretic properties to help characterize finite factorization properties of commutative rings, and conversely.

Article Details

How to Cite
Mooney, C. P. (2015). On irreducible divisor graphs in commutative rings with zero-divisors. Tamkang Journal of Mathematics, 46(4), 365–388. https://doi.org/10.5556/j.tkjm.46.2015.1753
Issue
Vol. 46 No. 4 (2015)
Section
Papers
Author Biography

Christopher Park Mooney, Westminster College

Visiting Assistant Professor,Department of Mathematics.

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