When a zero-divisor graph is planar or a complete r-partite graph

✍ Scribed by S. Akbari; H.R. Maimani; S. Yassemi

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 218 KB
• Volume: 270
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00370-3

✦ Synopsis

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 |Z(R)| = p 2 , |R| = p 3 , and R is isomorphic to exactly one of the rings Z p 3 , Z p [x,y] (xy,y 2 -x) , Z p 2 [y] (py,y 2 -ps)

, where 1 s < p.