Generalized Lie Ideal

h t t p : / / j o u r n a l s. t u b i t a k. g o v. t r / m a t h / Abstract: Let R be a *-prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero *-(σ, τ)-Lie ideal of R such that τ commutes with * , and a, b be in R. (i) If a ∈ S * (R) and [U, a] = 0 , then a ∈ Z (R) or U ⊂ Z (R). (ii) If a ∈ S * (R) and [U, a] σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R). (iii) If U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ , then there exists a nonzero *-ideal M of R such that [R, M ] σ,τ ⊂ U but [R, M ] σ,τ ̸ ⊂ Cσ,τ. (iv) Let U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ. If aU b = a * U b = 0 , then a = 0 or b = 0.

Abstract: Let R be a ∗ -prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero ∗ -(σ, τ)-Lie ideal of R such that τ commutes with ∗ , and a, b be in R. (i) If a ∈ S∗ (R) and [U, a] = 0, then a ∈ Z (R) or U ⊂ Z (R) . (ii) If a ∈ S∗ (R) and [U, a]σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R) . (iii) If U ̸⊂ Z (R) and U ̸⊂ Cσ,τ , then there exists a nonzero ∗ -ideal M of R such that [R,M ]σ,τ ⊂ U but [R,M ]σ,τ ̸⊂ Cσ,τ . (iv) Let U ̸⊂ Z (R) and U ̸⊂ Cσ,τ . If aUb = a∗Ub = 0, then a = 0 or b = 0.

h t t p : / / j o u r n a l s. t u b i t a k. g o v. t r / m a t h / Abstract: Let R be a *-prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero *-(σ, τ)-Lie ideal of R such that τ commutes with * , and a, b be in R. (i) If a ∈ S * (R) and [U, a] = 0 , then a ∈ Z (R) or U ⊂ Z (R). (ii) If a ∈ S * (R) and [U, a] σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R). (iii) If U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ , then there exists a nonzero *-ideal M of R such that [R, M ] σ,τ ⊂ U but [R, M ] σ,τ ̸ ⊂ Cσ,τ. (iv) Let U ̸ ⊂ Z (R) and U ̸ ⊂ Cσ,τ. If aU b = a * U b = 0 , then a = 0 or b = 0.

Let R be a prime ring with characteristic not two. U a (σ, τ)-left Lie ideal of R and d : R → R a non-zero derivation. The purpose of this paper is to invesitigate identities satisfied on prime rings. We prove the following results: (1) [d(R), a] = 0 ⇔ d([R, a]) = 0. (2) if (R, a) σ,τ = 0 then a ∈ Z. (3) if (R, a) σ,τ ⊂ C σ,τ then a ∈ Z. (4) if (U, a) ⊂ Z then a 2 ∈ Z or σ(u) + τ (u) ∈ Z, for all u ∈ U. (5) if (U, R) σ,τ ⊂ C σ,τ then U ⊂ Z.