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Paper IPM / M / 2323  


Abstract:  
Let Γ(R) be the zerodivisor graph of a commutative ring
R. An interesting question was proposed by Anderson, Frazier,
Lauve, and Livingston: For which finite commutative rings R is
Γ(R) planar? We give an answer to this question. More
precisely, we prove that if R is a local ring with at least 33
elements, and Γ(R) ≠ ∅, then Γ(R) is not
planar. We use the set of the associated primes to find the
minimal length of a cycle in Γ(R). Also, we determine the
rings whose zerodivisor graphs are complete rpartite graphs
and show that for any ring R and prime number p, p ≥ 3, if
Γ(R) is a finite complete ppartite graph, then
\TZ(R)=p^{2}, R=p^{3}, and R is isomorphic to exactly one of
the rings \mathbbZ_{p3}, [(\mathbbZ_{p}[x,y])/((xy,y^{2}−x))], [(\mathbbZ_{p2}[y])/((py,y^{2}−ps))],
where 1 ≤ s < p.
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