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On the Commutativity of a Ring with Identity

  • Jingcheng Tong (a1) (a2)

Abstract

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.

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References

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1. Bell, H. E., On the power map and ring commutativity, Canad. Math. Bull. 21 (1978), 399-404.
2. Bell, H. E., On rings with commuting powers, Math. Japon. 24 (1979/1980), 473-478.
3. Belluce, L. P., Herstein, I. N. and Jain, S. K., Generalized commutative rings, Nagoya Math. J. 27 (1966), 1-5.
4. Harmanci, A., Two elementary commutativity theorems for rings, Acta Math. Acad. Sci. Hungar. 29 (1977), 23-29.
5. Ligh, S. and Richoux, A., A commutativity theorem for rings, Bull. Austral. Math. Soc. 16 (1977), 75-77.
6. Luh, J., A commutativity theorem for primary rings, Acta Math. Acad. Sci. Hungar. 22 (1971), 211-213.
7. Nicholson, W. K. and Yaqub, A., A commutativity theorem for rings and groups, Canad. Math. Bull. 22 (1979), 419-423.
8. Nicholson, W. K. and Yaqub, A., A commutativity theorem, Algebra Universalis 10 (1980), 260-263.
9. Richoux, A., On a commutativity theorem of Luh, Acta Math. Acad. Sci. Hungar. 34 (1979), 23-25.
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On the Commutativity of a Ring with Identity

  • Jingcheng Tong (a1) (a2)

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