Let R be a noncommutative prime ring with its Utumi ring of quotients U , C = Z(U) the extended centroid of R, F a generalized derivation of R and I a nonzero ideal of R. Suppose that there exists 0 = a ∈ R such that a(F ([x, y]) − [x, y]) = 0 for all x, y ∈ I, where n ≥ 2 is a fixed integer. Then one of the following holds: 1. char (R) = 2, R ⊆ M2(C), F (x) = bx for all x ∈ R with a(b − 1) = 0...

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 $0neq a in R$. If $R$ admits a generalized derivation $F$ such that $a(F(u^2)pm F(u)^{2})=0$ for all $u in L$, then one of the following holds: begin{enumerate} item there exists $b in U$ such that $F(x)=bx$ for all $x in R$, with $ab=0$; item $F(x)=...

We prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison.

‎‎a natural generalization of two dimensional cyclic code ($t{tdc}$) is two dimensional skew cyclic code‎. ‎it is well-known that there is a correspondence between two dimensional skew cyclic codes and left ideals of the quotient ring $r_n:=f[x,y;rho,theta]/_l$‎. ‎in this paper we characterize the left ideals of the ring $r_n$ with two methods and find the generator matrix for two dimensional s...

For an arbitrary ring R, the largest strong left quotient ring Ql (R) of R and the strong left localization radical lR are introduced and their properties are studied in detail. In particular, it is proved that Ql (Q s l (R)) ≃ Q s l (R), l s R/ls R = 0 and a criterion is given for the ring Ql (R) to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring Q...

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An add...