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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A decomposition of elements of the free algebra
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by Wen Xin Ma
Proc. Amer. Math. Soc. 118 (1993), 37-45
DOI: https://doi.org/10.1090/S0002-9939-1993-1126198-4

Abstract:

Let $f$ be an element of $F[X]$, the free associative algebra over a field $F$ and $n$ the maximum of the degrees of the variables and the multiplicities of the degrees in $f$. A partial ordering on the homogeneous elements of $F[X]$ is defined such that if $f$ is homogeneous and $F\nmid n!$, then $f$ can be decomposed into a sum of two polynomials ${f_0}$ and ${f_1}$ such that for $0 < m \leqslant n,\;{f_0}$ is symmetric or skew symmetric in all its arguments of degree $m$ depending on whether $m$ is even or odd and ${f_1}$ is a consequence of polynomials of lower type than $f$. Osborn’s Theorem about the symmetry of the absolutely irreducible polynomial identities is obtained as a corollary. The same holds in the free nonassociative algebra. The proofs are combinatorial.
References
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Bibliographic Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 118 (1993), 37-45
  • MSC: Primary 16S10
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1126198-4
  • MathSciNet review: 1126198