In a recent paper, Vukman [12] gives an elementary but lengthy calculation that yields extensions of a well-known theorem of Posner [10] on centralizing derivations of prime rings. The purpose of this note is to show that by using the theory of differential identities one can fairly quickly generalize the results of Vukman to higher commutators, eliminate his restriction on characteristic, and extend the results from prime rings to Lie ideals in prime rings.The theory of differential identities was formally initiated by Kharchenko [5]. Its use here will not be explicit and arises only in reference to results in subsequent work of Chuang [1] and the author [6]. The one related object we need to mention is the symmetric quotient ring, introduced in [5] and required in [1] and [6] (see [9] for some details). Henceforth, we let R denote a prime ring with extended centroid C and symmetric quotient ring Q. All that we need here about these objects is that R c Q, Q is a prime ring whose center is the field C, and that C is the centralizer of ^ in Q. By D we always mean a nonzero derivation of R, and D = ad(;4) for A £ Q implies that We consider the more general Engel condition when for a fixed k > 0, [D(x), x]k = 0 for all x £ L, a noncommutative Lie ideal of R. Our first theorem will give the result for ideals. It incorporates the arguments needed for Lie ideals but avoids some technical complications of that case.Two well-known observations are crucial to our arguments and for convenience we state them as lemmas.

Two well-known results of E. C. Posner state that the composition of two nonzero derivations of a prime ring cannot be a nonzero derivation, and that in a prime ring, if the commutator of each element and its image under a nonzero derivation is central, then the ring is commutative. Our purpose is to show how the theory of differential identities can be used to obtain these results and their generalizations to Lie ideals and to rings with involution.

In this paper we study a question which, although somewhat special, has the virtue that its answer can be given in a very precise, definitive, and succinct way. It shows that the structure of a ring is very tightly determined by the imposition of a special behavior on one of its derivations.The problem which we shall examine is: Suppose that R is a ring with unit element, 1, and that d ≠ 0 is a derivation of R such that for every x ∊ R, d(x) = 0 or d(x) is invertible in R; must R then have a very special structure?As we shall see, the answer to this question is yes, in particular we show that except for a special case which occurs when 2R = 0, R must be a division ring D or the ring D2 of 2 × 2 matrices over a division ring.

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