SOME RESULTS ON (σ,τ)-DERIVATION IN PRIME RINGS

Abstract

Let R be a prime ring and d. R→R be a (o,t)-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 Rand 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.

(i) If UacZ(R) (or aUCZ(R)), for aeR), then 20 or R is commutative.

(ii) If d is a nonzero on R such that d(U)ac (R) (or ad(U)Z(R) for a∈Z(R), then either

0-0 or (1)+(U)Z(R).

(iii) If d is a nonzero homomorphism on U such that d(Uja Z(R)(or ad(U)Z(R))

for

GER, then a-0 or a()+(U)(R).