A Basis for Z-graded Identities of Matrices over Infinite Fields

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. Introduction. Let us denote by Mn(K) the algebra of all square matrices of order n over a field K. The polynomial identities of the algebra Mn(K) 2000 Mathematics Subject Classification: 16R10, 16R20, 16R50.