Proceedings of the American Mathematical Society

${\mathbf Z}_n$ -graded polynomial identities of the full matrix algebra of order $n$

Author(s): Sergei Yu. Vasilovsky
Journal: Proc. Amer. Math. Soc. 127 (1999), 3517-3524.
MSC (1991): Primary 16R40
Posted: May 13, 1999
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Abstract: The algebra

of all $n\times n$ matrices over a field

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

over a field of characteristic zero are studied. It is shown that all the $\mathbf{Z}_n$ -graded polynomial identities of

follow from the following:

$\begin{displaymath}x_1x_2-x_2x_1=0,~~~~\alpha (x_1)=\alpha (x_2)=\overline{0};\end{displaymath}$

$\begin{displaymath}x_1xx_2-x_2xx_1=0,~~~~\alpha (x_1)=\alpha (x_2)=-\alpha (x).\end{displaymath}$

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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

DOI: 10.1090/S0002-9939-99-04986-2
PII: S 0002-9939(99)04986-2
Keywords: Graded polynomial identities, full matrix algebra
Received by editor(s): April 10, 1997
Received by editor(s) in revised form: February 26, 1998
Posted: May 13, 1999
Communicated by: Ken Goodearl
Copyright of article: Copyright 1999, American Mathematical Society

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