Doctoral Dissertations

Date of Award


Degree Type


Degree Name

Doctor of Philosophy



Major Professor

David F. Anderson

Committee Members

Michael Langston, Shashikant Mulay, Pavlos Tzermias


Let R be a commutative ring with 1 ≠ 0, and let Z(R) denote the set of zero-divisors of R. One can associate with R a graph Γ(R) whose vertices are the nonzero zero-divisors of R. Two distinct vertices x and y are joined by an edge if and only if xy = 0 in R. Γ® is often called the zero-divisor graph of R. We determine which finite commutative rings yield a planar zero-divisor graph. Next, we investigate the structure of Γ(R) when Γ(R) is an infinite planar graph. Next, it is possible to extend the definition of the zero-divisor graph to a commutative semigroup. We investigate the problem of extending the definition of the zero-divisor graph to a noncommutative semigroup, and attempt to generalize results from the commutative ring setting. Finally, we investigate the structure of Γ(k1 × ∙ ∙ ∙ × kn) where each ki is a finite field. The appendices give planar embeddings of many families of zero-divisor graphs.

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