WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED

Year 2017, Volume: 21 Issue: 21, 198 - 216, 17.01.2017

Jesse Gerald Smith Jr

https://doi.org/10.24330/ieja.296332

Cited By: 1

Abstract

Let R be a commutative ring with nonzero identity and I a proper
ideal of R. The ideal-based zero-divisor graph of R with respect to the ideal
I, denoted by ΓI (R), is the graph on vertices {x ∈ R \ I | xy ∈ I for some
y ∈ R\I}, where distinct vertices x and y are adjacent if and only if xy ∈ I. In
this paper, we give a complete classification of when an ideal-based zero-divisor
graph of a commutative ring is complemented or uniquely complemented based
on the total quotient ring of R/I.

Keywords

: Zero-divisor graph, ideal-based, complemented graph, uniquely complemented graph

References

• [1] D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann
• regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221–241.
• [2] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative
• ring, J. Algebra, 217(2) (1999), 434–447.
• [3] I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
• [4] R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings,
• Comm. Algebra, 30(2) (2002), 745–750.
• [5] P. S. Livingston, Structure in Zero-Divisor Graphs of Commuative Rings, Masters
• Thesis, Univeristy of Tennessee, Knoxville, TN, 1997
• [6] S. P. Redmond, Generalizations of the Zero-Divisor Graph of a Ring, Ph.D.
• Thesis, The University of Tennessee, Knoxville, TN, 2001.
• [7] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring,
• Comm. Algebra, 31(9) (2003), 4425–4443.
• [8] J. G. Smith, Properties of Ideal-Based Zero-Divisor Graphs of Commutative
• Rings, Ph.D. Thesis, The Univeristy of Tennessee, Knoxville, TN, 2014.