Thesis Title

Planar Zero-Divisor Graphs

Date of Graduation

Summer 2006

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Richard Belshoff

Keywords

commutative ring, zero-divisor, zero-divisor graph, Artinian ring, local ring

Subject Categories

Mathematics

Abstract

Associated to every nonzero commutative ring with identity is a graph whose vertices are the nonzero zero-divisors, and such that two distinct vertices x and y are adjacent if and only if xy = 0. this graph is called the zero-divisor graph of the ring. Now assume that the ring is not a field. A result of Akbari, Maimani, and Yassemi (Journal of Algebra 270 (2003) 169-180) states that for any local ring with more than 32 elements, the zero-divisor graph is not a planar graph. In this thesis we show that for any local ring with more than 27 elements, the zero-divisor graph is not a planar graph. Moreover, we determine all finite local rings for which the zero-divisor graph is planar.

Copyright

© Jeremy M. Chapman

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