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| Paper IPM / M / 2323 |
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| Abstract: | |||||||
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Let Γ(R) be the zero-divisor 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 zero-divisor graphs are complete r-partite graphs
and show that for any ring R and prime number p, p ≥ 3, if
Γ(R) is a finite complete p-partite graph, then
|\TZ(R)|=p2, |R|=p3, and R is isomorphic to exactly one of
the rings \mathbbZp3, [(\mathbbZp[x,y])/((xy,y2−x))], [(\mathbbZp2[y])/((py,y2−ps))],
where 1 ≤ s < p.
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