Matrix Algebra Variety of Algebras Polynomial Identities Graded Identities
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 29, No 2, (2003), 149p-158p
The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an
infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities:
x = 0, |α(x)| ≥ n,
xy = yx, α(x) = α(y) = 0,
xyz = zyx, α(x) = −α(y) = α(z ),
where α is the degree of the corresponding variable.
This is a generalization of a result of Vasilovsky about the Z-graded identities of the algebra Mn(K) over fields of characteristic 0.