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On the Commutativity of a Ring with Identity

Published online by Cambridge University Press:  20 November 2018

Jingcheng Tong
Jingcheng Tong*
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA Institute of Mathematics, Academia Sinica, Peking, China
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Abstract

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Let R be a ring with identity. R satisfies one of the following properties for all x, y ∈ R:

  1. (I) xynxmy = xm+1yn+1 and mnm! n! x≠0 except x = 0;

  2. (II) xynxm = xm + 1yn + 1 and mm! n! x≠0 except x = 0;

  3. (III) xmyn = ynxm and m! n! x≠0 except x = 0;

  4. (IV) (xpyQ)n = xpnyqn for n = k, k + 1 and N(p, q, k) x≠0 except x = 0, where N(p, q, k) is a definite positive integer. Then R is commutative.

Keywords

16A70
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 4 , 01 December 1984 , pp. 456 - 460
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Bell, H. E., On the power map and ring commutativity, Canad. Math. Bull. 21 (1978), 399-404.10.4153/CMB-1978-070-xCrossRefGoogle Scholar
2. Bell, H. E., On rings with commuting powers, Math. Japon. 24 (1979/1980), 473-478.Google Scholar
3. Belluce, L. P., Herstein, I. N. and Jain, S. K., Generalized commutative rings, Nagoya Math. J. 27 (1966), 1-5.CrossRefGoogle Scholar
4. Harmanci, A., Two elementary commutativity theorems for rings, Acta Math. Acad. Sci. Hungar. 29 (1977), 23-29.CrossRefGoogle Scholar
5. Ligh, S. and Richoux, A., A commutativity theorem for rings, Bull. Austral. Math. Soc. 16 (1977), 75-77.10.1017/S0004972700023029CrossRefGoogle Scholar
6. Luh, J., A commutativity theorem for primary rings, Acta Math. Acad. Sci. Hungar. 22 (1971), 211-213.10.1007/BF01896012CrossRefGoogle Scholar
7. Nicholson, W. K. and Yaqub, A., A commutativity theorem for rings and groups, Canad. Math. Bull. 22 (1979), 419-423.10.4153/CMB-1979-055-9CrossRefGoogle Scholar
8. Nicholson, W. K. and Yaqub, A., A commutativity theorem, Algebra Universalis 10 (1980), 260-263.10.1007/BF02482908CrossRefGoogle Scholar
9. Richoux, A., On a commutativity theorem of Luh, Acta Math. Acad. Sci. Hungar. 34 (1979), 23-25.10.1007/BF01902588CrossRefGoogle Scholar