Some results on (σ, τ)‎-derivation in prime rings

Abstract EN

Let R be a prime ring and d : R → R be a (σ, τ)-derivation of R.U be a left ideal of R which is semiprime as a ring.

In this paper we proved that if d is a nonzero endomorphism on R, and d (R) ⊂ Z (R), then R is commutative, and we show by an example the condition d is an endomorphism on R can not be excluded.

Also, we proved the following.

If Ua ⊂ Z (R) (or aU ⊂ Z (R)), for a ∈ R), then a = 0 or R is commutative, If d is a nonzero on R such that d (U) a ⊂ Z (R) (or ad (U) ⊂ Z (R) for a ∈ Z (R), then either a = 0 or σ (U) + τ (U) ⊂ Z (R).

If d is a nonzero homomorphism on U such that d (U) a ⊂ Z (R) (or ad (U) ⊂ Z (R)) for a ∈ R, then a = 0 or σ (U) + τ (U) ⊂ Z (R).