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1
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.
2
On (m, n)-Jordan derivations and commutativity of prime rings
3
An identity with derivations on rings and Banach algebras
4
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.
5
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.
6
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].
7
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.
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