prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements x and y of R are adjacent in the graph if and only if xRy = 0 or yRx = 0. prime graph of a ring is denoted by PG(R). Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.

Let $R$ be a prime ring of characteristic different from‎ ‎$2$ and $3$‎, ‎$Q_r$ its right Martindale quotient ring and‎ ‎$C$ its the extended centroid‎. ‎Suppose that $F$ is a non-zero generalized skew derivation of $R$ such that‎ ‎$F([x,y]_k)=[F(x),y]_k+[x,F(y)]_k$‎ ‎for all $x,y\in R$‎, ‎with $k>1$ fixed integer‎. ‎In this paper we will showw that‎, ‎then $R$ is commutative‎. $$F([x; y]_k) = [F(x); y]_k + [x; F(y)]_k$$ for all $x, y \in R$, with $k > 1$ fixed integer. Then $R$ is commutative.

Let R be a prime ring with extended centroid C, H a generalized derivation of R and n³1 a fixed integer. In this paper we study the situations: (1) If (H (xy)) n= (H (x)) n (H (y)) n for all x, y Î R; (2) obtain some related result in case R is a noncommutative Banach algebra and H is continuous or spectrally bounded.

Let R be a ring with derivation d, such that (d (xy)) n= (d (x)) n (d (y)) n for all x, y Î R and n³1 a fixed integer. In this paper, we show that if R is prime, then d=0 or R is commutative. If R is semiprime, thend maps R into its center. Moreover in semiprime case let A=O (R) be the orthogonal completion of R and B=B (C) be the Boolian ring of C, where C is the extended centroid of R. Then there exists an idempotent e Î B such that e A is a commutative ring and d induces a zero derivation on (1-e) A.

Let R be a prime ring with its Utumi ring of quotients U, C=Z (U) the extended centroid of R, L a non-central Lie ideal of R and 0 ¹aÎR. If R admits a generalized derivation F such that a (F (u2)±F(u) 2) =0 for all uÎL, then one of the following holds: (1) there exists bÎU such that F (x) =bx for all xÎR, with ab=0;(2)F (x)=±x for all xÎR; (3) char (R)=2 and R satisfies s4; (4) char (R)¹2, R satisfies s4 and there exists bÎU such that F (x) = bx for all xÎR.We also study the situations (i)a (F (xmyn)±F (xm) F (yn)) =0 for all x, yÎR, and (ii) a (F (xmyn)±F (yn) F (xm))=0 for all x, y ÎR in prime and semiprime rings.