关键词: Prime rings, Generalized (α,α)-derivations, Involution, Commutativity

Abstract:  Let R be a prime ring of characteristic different from two with the sec-
ond involution ∗ and α an automorphism of R . An additive mapping F of R is called
a generalized ( α,α )-derivation on R if there exists an ( α,α )-derivation d of R such
that F ( xy )= F ( x ) α ( y )+ α ( x ) d ( y ) holds for all x,y∈R. The paper deals with the s-
tudy of some commutativity criteria for prime rings with involution. Precisely, we
describe the structure of R admitting a generalized ( α,α )-derivation F satisfying any
one of the following properties: ( i ) F ( xx^∗) −α ( xx^∗) ∈Z ( R ). ( ii ) F ( xx^∗ )+ α ( xx^∗ ) ∈
Z ( R ). ( iii ) F ( x ) F ( xx^∗) −α ( xx^∗) ∈Z ( R ). ( iv ) F ( x ) F (x^∗)+ α ( xx^∗) ∈Z ( R ). ( v ) F ( xx^∗) −
F ( x ) F (x^∗ ) ∈Z ( R ). ( vi ) F ( xx^∗) −F (x^∗) F ( x )=0 for all x∈R . Also, some examples are
given to demonstrate that the restriction of the various results is not superfluous. In fact,
our results unify and extend several well known theorems in literature.

Key words: Prime rings, Generalized (α,α)-derivations, Involution, Commutativity