Planar Zero-Divisor Graphs
Date of Graduation
Master of Science in Mathematics
commutative ring, zero-divisor, zero-divisor graph, Artinian ring, local ring
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.
© Jeremy M. Chapman
Chapman, Jeremy M., "Planar Zero-Divisor Graphs" (2006). MSU Graduate Theses. 2315.