COMMUTING MAPPINGS ON RIGHT IDEALS IN PRIME RINGS

Abstract

Let R be a prime ring of characteristic different from 2, with extended centroid C, d and g derivations of R, I a non-zero right ideal of R and s4 the standard identity of degree 4. If [d([x,y]),[x,y]][x,y]− [x,y][g([x,y]),[x,y]] = 0, for all x,y ∈ I, then one of the following holds: (i) s4(x1,x2,x3,x4)x5 is an identity for I; (ii) d(x)=[a,x], with (a−α)I = 0 for a suitable α ∈ C and g =0 . Let R be a prime ring with center Z(R) and extended centroid C, Q its Martindale quotient ring. Here we will consider some related problems concerning derivations in prime rings which satisfy some commuting conditions. Our aim is to study the relationship between the behaviour of such derivations and the structure of R. Recall that a mapping F from R to R is said to be commuting on R if [F(x),x] = 0, for allx ∈ R, and is said to be centralizing on R if [F(x),x ] ∈Z (R), for all x ∈ R. There has been considerable interest in commuting, centralizing and related mappingsin prime and semiprime rings (see for istance [2]). In[11]Posner provedthat the existence of a noncentralizing derivation d on a prime ring R, forces R to be commutative. Later in [12] Vukman has proved that in case there exists a non-zero derivation d on R, whereR is a prime ring of characteristic different from 2 and 3, such that the mapping x −→ [d(x),x ] is centralizing on R, thenR is commutative. In a recent paper [7] Jun and Kim proved that if d(x)x− xg(x) ∈ Z(R), for d and g derivations of R and