Nilpotency of derivations in prime rings

David W. Jensen
1995 Proceedings of the American Mathematical Society  
In 1957, E. C. Posner proved that if X and ô are derivations of a prime ring R, characteristic R / 2, then XS = 0 implies either X = 0 or S = 0. We extend this well-known result by showing that, without any characteristic restriction, Xôm = 0 implies either X = 0 or <54m_1 = 0. We also prove that XnS = 0 implies either S2 = 0 or A12"-9 = 0. In the case where X"ôm = 0 , we show that if X and ô commute, then at least one of the derivations must be nilpotent. A derivation of a ring 7? is an
more » ... e map X: R -> R satisfying X(xy) = Xx • y + x • Xy for all x, y £ R. In 1957, Edward Posner showed that if X and ô are derivations of a prime ring R, characteristic R / 2, then XS = 0 implies either X = 0 or ô = 0 (see [3] ). In this paper we show that in a prime ring with no characteristic restriction, Xô = 0 implies either X = 0 or ô2 = 0. We generalize this result by proving that for any positive integer m , Xôm = 0 implies either X = 0 or J4"1-1 = 0. Furthermore, we show that for any positive integer n , X"S = 0 implies either Ô2 = 0 or Xx2"~9 = 0. Lastly, in the general case where Xnôm = 0, we prove that if X and ô commute, then at least one of the derivations must be nilpotent. Lemma 1. Assume X is a derivation of a prime ring R and 3a £ R, a ^ 0, such that a(XnR) = 0 or (XnR)a = 0 Then X2"-x =0. Proof. Start by assuming a(XnR) = 0. Then for all x, y £ R, we have (1) aXn(xy) = a I¿ (n\xixX"-iy\ = 0. Note that replacing x by X"~xx in (1) yields aX"~xxX"y = 0. Replacing x by X"~2x and y by Xy in (1), and using the fact that aXn~xxX"y = 0, we get aX"~2xXn+xy = 0. At the next iteration we replace x by Xn"ix and y by X2y in (1) , and use aXn~xxXny = 0 and aX"-2xXn+xy = 0, to get aXn^xXn+2y = 0. Continuing this process we eventually obtain axX2n~xy = 0. Since this is true for all x and y in 7?, and 7? is a prime ring, we conclude that X2n~x = 0. Similarly, if we begin with (XnR)a = 0, we reach the same conclusion. D
doi:10.1090/s0002-9939-1995-1291775-9 fatcat:hqci2hd7yjbjvf6pen2w2ca3zm