## Journal of Commutative Algebra

### Invariants and isomorphism theorems for zero-divisor graphs of commutative rings of quotients

John D. LaGrange

#### 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.

#### Article information

Source
J. Commut. Algebra, Volume 6, Number 3 (2014), 407-437.

Dates
First available in Project Euclid: 17 November 2014

https://projecteuclid.org/euclid.jca/1416233325

Digital Object Identifier
doi:10.1216/JCA-2014-6-3-407

Mathematical Reviews number (MathSciNet)
MR3278811

Zentralblatt MATH identifier
1308.13011

#### Citation

LaGrange, John D. Invariants and isomorphism theorems for zero-divisor graphs of commutative rings of quotients. J. Commut. Algebra 6 (2014), no. 3, 407--437. doi:10.1216/JCA-2014-6-3-407. https://projecteuclid.org/euclid.jca/1416233325

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