On Centralizing Automorphisms and Jordan Left Derivations on sigma-Prime Gamma Rings

Derivations on Lie Ideals of σ-Prime Γ-Rings

“…They described additive mappings d : R  R such that d(u 2 ) = 2u d(u)u  U, where U is a nonzero -square closed Lie ideal of a 2-torsion free -prime ring R and proved that d(uv ) = ud(v) + vd(u) u, v  U. also studied Jordan generalized derivations of -prime rings and proved that every Jordan generalized derivations on U of R is a generalized derivations on U of R, where U is a -square closed Lie ideal of a 2-torsion free -prime ring R. Some significant results developed on Lie ideals and generalized derivations in -prime rings by Khan and Khan (2012). Some characterizations of centralizing automorphisms on a -square closed Lie ideals of -prime -rings have been developed by Dey et al (2015). They studied Jordan left derivations on a -square closed Lie ideals and proved that such type of Jordan derivations is a derivation on a -square closed Lie ideals of a -prime -ring.…”

“…An example of an involution and an example of a -prime -ring which is not a prime -ring appeared in Dey and Paul (2015). On the other hand, various remarkable characterizations of -prime rings on -square closed Lie ideals have been studied by many authors viz.…”

Derivations Acting as Homomorphisms and as Anti-homomorphisms in σ-Lie Ideals of σ-Prime Gamma Rings

“…[8], Lemma 3.1) Let U ̸ = 0 be a σ-ideal of a 2-torsion free σ-prime Γ-ring M satisfying the condition (*). If[U, U ] Γ = 0, then U ⊆ Z(M ).Lemma 2.2 ([7], Lemma 2.2) Let U Z(M ) be a σ-ideal of a 2-torsion free σ-prime Γ-ring M satisfying the condition (*) and a, b ∈ M such that aαU βb = aαU βσ(b) = 0 for all α, β ∈ Γ.…”