Vol. 99, No. 2, 1982

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Derivations of higher order and commutativity of rings

Lung O. Chung and Jiang Luh

Vol. 99 (1982), No. 2, 317–326

Let R be an associative ring with center C, and be a derivation on R. The authors consider the commutativity of R which satisfies the property that nx mx C, where n > m are fixed nonnegative integers. An example is given to show that if m 2, R may not be commutative. For 0 m 2, suppose R is either r-torsion free with large r or torsion free. It is shown that (i) if nx ± x C for all x R then all commutators of R are central; (ii) if nx ± ∂x C for all x R and n is even then ∂x∂y ∂y∂x C for all x,y R; (iii) if nx ± 2x C for all x R and if n is odd then 2x∂2y 2y∂2x C for all x,y R. In all these cases, if one assumes further that R is prime, then must be trivial. Examples are also given to illustrate that some of these assumptions on evenness of n, and that r being large are essential. Finally, those integral domains which have nx central for all x are also studied. They are shown to be commutative.

Mathematical Subject Classification
Primary: 16A70, 16A70
Received: 17 November 1980
Revised: 21 May 1981
Published: 1 April 1982
Lung O. Chung
Jiang Luh