SOME RESULTS ON (σ;tau)-DERIVATION IN PRIME RINGS

Let R be a prime ring and d: R ~R be a (σ;tau)-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)subset 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 fol lowing. (i) If Uasubset Z(R) (or aUsubset Z(R)), for aє R ), then a=O or R is commutative. (ii) If dis a nonzero on R such that d(U)asubset Z(R) (or ad(U)subset Z(R) for aєZ(R), then either a=O or σ( U)+tau( U)subset Z(R) . (iii) If dis a nonzero homomorphism on U such that d(U)asubset Z(R)(or ad(U)subset Z(R)) for aeR, then a=o or σ(U)+tau(U)subset Z(R).