Vol. 104, No. 1, 1983

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On the iterates of derivations of prime rings

Wallace Smith Martindale, III and C. Robert Miers

Vol. 104 (1983), No. 1, 179–190

In this paper we study properties of associative derivations whose iterates are related in rather special ways to the original derivation, or to the iterates of another derivation. An associative derivation d : R R is an additive (or linear when appropriate) mapping on a ring R satisfying d(xy) = xd(y) + d(x)y for all x,y R. A derivation d : R R is called inner if d(x) = (ad a)(x) for some a R where (ad a)(x) = [a,x] = ax xa. In particular we ask when can the iterate of an inner derivation be an inner derivation? When can the iterates of two derivations commute? More precisely, we characterize elements a,b R, R a prime ring, for which (ad a)n(x) = (ad b)(x) for all x R, and we characterize derivations d : R R, δ : R R for which [dn(x)n(y)] = 0 for all x,y R, R prime. Applications are made to C-algebras.

Mathematical Subject Classification 2000
Primary: 16A72, 16A72
Secondary: 46L05, 16A12
Received: 31 October 1980
Revised: 10 February 1982
Published: 1 January 1983
Wallace Smith Martindale, III
C. Robert Miers