Vol. 99, No. 2, 1982

Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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