Cliques in the extended zero-divisor graph of finite commutative rings.

Let R be a finite commutative ring with or without unity and Γe(R) be its extended zero-divisor graph with vertex set Z*(R) = Z(R)\{0} and two distinct vertices x, y are adjacent if and only if x.y = 0 or x + y ∈ Z*(R). In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number 1 or 2. We completely characterize the rings of the form R ≅ R1 x R2, where R1 and R2 are local, having clique number 3, 4 or 5. Further we determine the rings of the form R ≅ R1 x R2 x R3, where R1, R2 and R3 are local rings, to have clique number equal to six. [ABSTRACT FROM AUTHOR]