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Proceedings of the American Mathematical Society

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Rings satisfying monomial identities

Authors: Mohan S. Putcha and Adil Yaqub
Journal: Proc. Amer. Math. Soc. 32 (1972), 52-56
MSC: Primary 16.49
MathSciNet review: 0289569
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Abstract: The following theorem is proved: Suppose R is an associative ring and suppose that $ w({x_1}, \cdots ,{x_n})$ is a fixed word distinct from $ {x_1} \cdots {x_n}$. If, further, $ {x_1} \cdots {x_n} = w({x_1}, \cdots ,{x_n})$, for all $ {x_1}, \cdots ,{x_n}$ in R, then the commutator ideal of R is nilpotent. Moreover, it is shown that this theorem need not be true if the word w is not fixed.

References [Enhancements On Off] (What's this?)

  • [1] I. N. Herstein, Theory of rings, Math. Lecture Notes, University of Chicago, Chicago, Ill., 1961.
  • [2] N. Jacobson, Structure of rings, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1964. MR 36 #5158. MR 0222106 (36:5158)
  • [3] N. H. McCoy, The theory of rings, Macmillan, New York; Collier-Macmillan, London, 1964. MR 32 #5680. MR 0188241 (32:5680)
  • [4] M. S. Putcha and A. Yaqub, Semigroups satisfying permutation identities, Semigroup Forum 13 (1971), 68-73. MR 0292969 (45:2050)

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