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FALL 2014 Invariants and isomorphism theorems for zero-divisor graphs of commutative rings of quotients
John D. LaGrange
J. Commut. Algebra 6(3): 407-437 (FALL 2014). DOI: 10.1216/JCA-2014-6-3-407

Abstract

Given a commutative ring $R$ with $1\neq0$, the zero-divisor graph $\Gamma(R)$ of $R$ is the graph whose vertices are the nonzero zero-divisors of $R$, such that distinct vertices are adjacent if and only if their product in $R$ is $0$. It is well known that the zero-divisor graph of any ring is isomorphic to that of its total quotient ring. This result fails for more general rings of quotients. In this paper, conditions are given for determining whether the zero-divisor graph of a ring of quotients of $R$ is isomorphic to that of $R$. Examples involving zero-divisor graphs of rationally $\aleph_0$-complete commutative rings are studied extensively. Moreover, several graph invariants are studied and applied in this investigation.

Citation

Download Citation

John D. LaGrange. "Invariants and isomorphism theorems for zero-divisor graphs of commutative rings of quotients." J. Commut. Algebra 6 (3) 407 - 437, FALL 2014. https://doi.org/10.1216/JCA-2014-6-3-407

Information

Published: FALL 2014
First available in Project Euclid: 17 November 2014

zbMATH: 1308.13011
MathSciNet: MR3278811
Digital Object Identifier: 10.1216/JCA-2014-6-3-407

Subjects:
Primary: 05C25 , 13A99

Keywords: ring of quotients , zero-divisor graph

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.6 • No. 3 • FALL 2014
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