On generalized derivations satisfying certain identities

Left Annihilator of Identities Involving Generalized Derivations in Prime Rings

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)=...

Derivations on Certain Semigroup Algebras

In the present paper we give a partially negative answer to a conjecture of Ghahramani, Runde and Willis. We also discuss the derivation problem for both foundation semigroup algebras and Clifford semigroup algebras. In particular, we prove that if S is a topological Clifford semigroup for which Es is finite, then H1(M(S),M(S))={0}.

Generalized Derivations on Modules

Generalized Derivations on Modules *

Let A be a Banach algebra and M be a Banach left A-module. A linear map δ : M → M is called a generalized derivation if there exists a derivation d : A → A such that δ(ax) = aδ(x) + d(a)x (a ∈ A,x ∈ M). In this paper, we associate a triangular Banach algebra T to Banach A-module M and investigate the relation between generalized derivations on M and derivations on T . In particular, we prove th...

On (anti-)multiplicative Generalized Derivations

Let R be a semiprime ring and let F, f : R → R be (not necessarily additive) maps satisfying F (xy) = F (x)y + xf(y) for all x, y ∈ R. Suppose that there are integers m and n such that F (uv) = mF (u)F (v) + nF (v)F (u) for all u, v in some nonzero ideal I of R. Under some mild assumptions on R, we prove that there exists c ∈ C(I) such that c = (m + n)c2, nc[I, I] = 0 and F (x) = cx for all x ∈...