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1
Remarks on *-(σ, τ)-Lie ideals of *-prime rings with derivation
EN
Let R be a *- prime ring with characteristic not 2, U a nonzero *- (σ, τ)-Lie ideal of R, d a nonzero derivation of R. Suppose σ, τ be two automorphisms of R such that σd = dσ, τd = dτ and * commutes with σ, τ, d. In the present paper it is shown that if d(U) ⊆ Z or d2 (U) ⊆ Z, then U ⊆ Z.
2
On derivations of operator algebras with involution
EN
The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) (…) L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A (…) A(X). In this case, D is of the form D(A) = [A,B] for all A (…) A(X) and some fixed B (…) L(X), which means that D is a derivation.
3
Identities with generalized derivations in semiprime rings
EN
Let R be a semiprime ring. An additive mapping F:R  R is called a generalized derivation of R if there exists a derivation d : R  R such that F(xy) = F(x)y + xd(y) holds, for all x,y  R. The objective of the present paper is to study the following situations: (1) (...), for all x, y in some appropriate subset of R.
4
A note on generalized (m, n)-Jordan centralizers
EN
The aim of this paper is to define generalized (m, n)-Jordan centralizers and to prove that on a prime ring with nonzero center and char (R) ≠ 6mn(m+n)(m+2n) every generalized (m, n)-Jordan centralizer is a two-sided centralizer.
5
Pairs of derivations on rings and Banach algebras
EN
We give a generalization of Vukman's theorem concerning a pair of derivations on rings. Then applying this purely algebraic result we obtain several range inclusion results of pair of derivations on Banach algebras.
6
Jordan structure on prime rings with centralizers
EN
Our object in this paper is to study the generalization of Borut Zalar result in [1] on Jordan centralizer of semiprime rings by prove the following result: Let R be a prime of characteristic different from 2, and U be a Jordan ideal of R. If T is an additive mapping from R to itself satisfying the following condition T(ur + ru) = uT(r) + T(r)u, then T(ur) = uT(r), for all r is an element of R, u is an element of U.
7
Identities with two automorphisms on semiprime rings
EN
In this paper we investigate identities with two automorphisms on semiprime rings. We prove the following result: Let T, S : R approaches R be automorphisms where R is a 2-torsion free semiprime ring satisfying the relation T(x)x = xS(x) for all x is an element of R. In this case the mapping x approaches T(x) - x maps R into its center and T = S.
8
On some equations related to derivations in rings and Banach algebras
EN
The main purpose of this paper is to investigate additive mapping D : R -> R, where R is a (m + n +1)! and \m2 + n2 - m - n - 4mn\ -torsion free semiprime ring with the identity element, satisfying the relation 2D(xm+n+l) = (m+-n+1)(xmD(x)xn +-xnD(x)xm), for all is an element of R and some integers m > 1, n > 1, m2 + n2 - m - n - 4mn /=0.
9
Identities with products of (alpha, beta)-derivations on prime rings
EN
The main purpose of this paper is to prove the following result. Let R be a noncommutative prime ring of characteristic different from two and let D and G = 0 be (\alpha, beta)-derivations of R into itself such that G commutes with alpha and beta. If [D{x), G(x)] = 0 holds for all x is an eleemnt of R then D = lambdaG where lambda is an element from the extended centroid of R.
10
On alfa-derivations of prime and semiprime rings
EN
In this paper we investigate identities with alfa-derivations on prime and semiprime rings. We prove, for example, the following result. If D : R - R is an alfa-derivation of a 2 and 3-torsion free semiprime ring R such that [D(x},x2] = 0 holds, for all x is an element of R, then D maps R into its center. The results of this paper are motivated by the work of Thaheem and Samman [20].
11
Free actions of semiprime rings with involution induced a derivation
EN
Let R be an associative ring. An element a is an element of R is said to be dependent of a mapping F : R -> R in case F (x) a = ax holds for all x is an element of R. A mapping F : R -> R is called a free action in case zero is the only dependent element of F. In this paper free actions of semiprime *- rings induced by a derivation are considered. We prove, for example, that in case we have a derivation D : R -> R, where R is a semiprime *-ring, then the mapping F defined by F(x) = D(x*) + D(x)*,x is an element of R, is a free action. It is also proved that any Jordan *-derivation on a 2-torsion free semiprime *-ring is a free action.
12
A note centralizers in semiprime rings
EN
The purpose of this paper is to prove the following result: Let R be a (m+n + 2)! and 3m2n + 3mn2 + 4m2 + 4n2 +10mn-torsion free semiprime ring with an identity element and let T : R -R be an additive mapping such that 3T(xm+n+1) = T(x)xm+n + xmT(x)xn + xm=nT(x) is fulfilled for all x is an element R and some fixed nonnegative integers m and n, m+n=0. In this case T is a centralizer.
13
On (alpha,beta)-derivations of semiprime rings
EN
We show that if alpha and beta are centralizing automorphisms and d a centralizing (alpha,beta)-derivation of a semiprime ring -R, then d is commuting. Some results on alpha-derivations and centralizing derivations of semiprime rings follow as applications of this result.
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