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A Commutattvity Theorem for Rings and Groups

Published online by Cambridge University Press:  20 November 2018

W. K. Nicholson  and
Adil Yaqub
W. K. Nicholson
Affiliation:
Department of Mathematics, The University of Calgary, Calgary, Alberta T2N 1N4
Adil Yaqub
Affiliation:
Department of Mathematics, University of California, Santa Barbara, Califorina 93106
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Abstract

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The following theorem is proved: Suppose R is a ring with identity which satisfies the identities xkyk = ykxk and xlyl = ylxl, where k and l are positive relatively prime integers. Then R is commutative. This theorem also holds for a group G. Furthermore, examples are given which show that neither R nor G need be commutative if either of the above identities is dropped. The proof of the commutativity of R uses the fact that G is commutative, where G is taken to be the group R* of units in R.

Keywords

16A7020F1016A38
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 4 , 01 December 1979 , pp. 419 - 423
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Jacobson, N., Structure of rings, A.M.S. Colloq. Publ., 37 (1964).Google Scholar