Abstract
LetR be an associative ring with identity which satisfies the identities(xy) k=(yx) k and(xy) l=(yx) l, wherek andl are relatively prime positive integers, depending onx andy. ThenR is commutative. Moreover, examples are given which show thatR need not be commutative if either of the above identities is dropped. This theorem is also true for groups.
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References
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D. L. Outcalt andAdil Yaqub,A commutativity theorem for power-associative rings, Bull. Austral. Math. Soc.3, (1970), 75–79.
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Nicholson, W.K., Yaqub, A. A commutativity theorem. Algebra Universalis 10, 260–263 (1980). https://doi.org/10.1007/BF02482908
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DOI: https://doi.org/10.1007/BF02482908