On Centralizing and Generalized Derivations Of prime Rings with involution

 Let (R,∗) be a 2-torsion free ∗-prime ring with involution ∗, L= 0 be a nonzero square closed ∗-Lie ideal of R and Z the center of R. An additive mapping F: R −→ R is called a generalized derivation on R if there exists a derivation d: R−→Rcommutes with ∗ such that F(xy) = F(x)y +xd(y) holds for all x,y ∈ R. In the present paper, we shall show that L is contained in the center of R such that R admits a generalized derivations F and G with associated derivations d and g commute with ∗ satisfying several conditions

Morita context and Generalized (α, β)−Derivations

Let $R$ and $S$ be rings of a semi-projective Morita context, and $alpha, eta$ be automorphisms of $R$. An additive mapping $F$: $Ro R$ is called a generalized $(alpha,eta)$-derivation on $R$ if there exists an $(alpha,eta)$-derivation $d$: $Ro R$ such that$F(xy)=F(x)alpha(y)+eta(x)d(y)$ holds for all $x,y in R$. For any $x,y in R$, set $[x, y]_{alpha, eta} = x alpha(y) - eta(y) x$ and $(x circ y)_{alpha, eta} = x alpha(y) + eta(y) x$. In the present paper, we shall show that if the ring $S$ is reduced then it is a commutative, in a compatible way with the ring $R$ . Also, we obtain some results on bialgebras via Cauchy modules