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Let R be a commutative ring and Z (R) be its zero-divisors set. The zero-divisor graph of R , denoted by Γ (R) , is an undirected graph with vertex set Z (R) ∗ = Z (R) ∖ { 0 } and two distinct vertices a and b are adjacent if and only if a b = 0. In this paper, for n = p R q S where p and q are primes (p < q) and R and S are positive integers, we calculate the A α spectrum of the graphs Γ (ℤ n). [ABSTRACT FROM AUTHOR]

Our intention in this paper is to prove the following. Let R be a ring with an idempotent element (0; 1 -)e and f be a multiplicative b-generalized derivation on R. Then we show that f is additive by imposing certain conditions on the ring R. [ABSTRACT FROM AUTHOR]

Let R be a 2-torsion free ring and U be a square closed Lie ideal of R. Suppose that α, β are automorphisms of R. An additive mapping δ: R → R is said to be a Jordan left (α,β)-derivation of R if δ(x²) = α(x)δ(x)+β(x)δ(x) holds for all x ∈ R. In this paper it is established that if R admits an additive mapping G : R→ R satisfying G(u²) = α(u)G(u)+α(u)δ(u) for all u ∈ U and a Jordan left (α,α)-derivation δ; and U has a commutator which is not a left zero divisor, then G(uv) = α(u)G(v)+α(v)δ(u) for all u, v ∈ U. Finally, in the case of prime ring R it is proved that if G: R → R is an additive mapping satisfying G(xy) = α(x)G(y)+β(y)δ(x) for all x, y ∈ R and a left (α, β)-derivation δ of R such that G also acts as a homomorphism or as an linebreak anti-homomorphism on a nonzero ideal I of R, then either R is commutative or δ = 0 on R. [ABSTRACT FROM AUTHOR]