Matrix identities with forms

Consider the algebra Mn(F) of n×n matrices over an infinite field F of arbitrary characteristic. An identity for Mn(F) with forms is such a polynomial in n×n generic matrices and in σk(x), 1≤k≤n, coefficients in the characteristic polynomial of monomials in generic matrices, that is equal to zero matrix. This notion is a characteristic free analogue of identities for Mn(F) with trace and it can be applied to the problem of investigation of identities for Mn(F). In 1996 Zubkov established an infinite generating set for the -idealTn of identities for Mn(F) with forms. Namely, for t>n he introduced partial linearizations of σt and proved that they together with the well-known free relations and the Cayley–Hamilton polynomial χn generate Tn as a -ideal. We show that it is enough to take partial linearizations of σt for nn, the well-known free relations, χt,r for t+2r=n, and ζt,r for t+2r=n−1, where σt,r is the identity introduced by Zubkov in 2005 and χt,r, ζt,r are generalizations of the Cayley–Hamilton polynomial. We prove that it is enough to take partial linearizations of σt,r for n