## $\mathbf {Z}_n$-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: \[ x_1x_2-x_2x_1=0,~~~~\alpha (x_1)=\alpha (x_2)=\overline {0};\] \[ x_1xx_2-x_2xx_1=0,~~~~\alpha (x_1)=\alpha (x_2)=-\alpha (x).\]## References

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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