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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$\mathbf {Z}_n$-graded polynomial identities of the full matrix algebra of order $n$
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by Sergei Yu. Vasilovsky PDF
Proc. Amer. Math. Soc. 127 (1999), 3517-3524 Request permission

Abstract:

The algebra $M_n(F)$ of all $n\times n$ matrices over a field $F$ has a natural $\mathbf {Z}_n$-grading $M_n(F)=\sum _{\alpha \in \mathbf {Z}_n}\bigoplus \mathcal {M}_n^{(\alpha )}$. In this paper graded identities of the $\mathbf {Z}_n$-graded algebra $M_n(F)$ over a field of characteristic zero are studied. It is shown that all the $\mathbf {Z}_n$-graded polynomial identities of $M_n(F)$ follow from the following: \[ x_1x_2-x_2x_1=0,~~~~\alpha (x_1)=\alpha (x_2)=\overline {0};\] \[ x_1xx_2-x_2xx_1=0,~~~~\alpha (x_1)=\alpha (x_2)=-\alpha (x).\]
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Additional Information
  • Sergei Yu. Vasilovsky
  • Affiliation: Department of Mathematics, University of Swaziland, Private Bag 4, Kwaluseni, Swaziland, Southern Africa; Institute of Mathematics, Novosibirsk 630090, Russia
  • Email: vasilovs@realnet.co.sz
  • Received by editor(s): April 10, 1997
  • Received by editor(s) in revised form: February 26, 1998
  • Published electronically: May 13, 1999
  • Communicated by: Ken Goodearl
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3517-3524
  • MSC (1991): Primary 16R40
  • DOI: https://doi.org/10.1090/S0002-9939-99-04986-2
  • MathSciNet review: 1616581