Generators of the T-ideal of polynomial identities for M_n(F) when n ≥ 3

Determine a generating set for the T-ideal of polynomial identities Id(M_n(F)) of the algebra M_n(F) of n×n matrices over a field F for all n ≥ 3, extending the known description available for n = 2.

Background

In discussing generators for T-ideals of PI algebras, the paper highlights that explicit descriptions are rare. A classical example with a known description is the algebra of upper triangular matrices UT_n(F) in characteristic zero, whose T-ideal is generated by the product of commutators u_n as shown by Mal’cev.

The authors note that even for the fundamental case of full matrix algebras M_n(F), despite the Amitsur–Levitzki theorem establishing that the standard polynomial St_{2n} is an identity, an explicit generating set for Id(M_n(F)) is known only for n = 2 (described by Drensky), and remains unknown for n ≥ 3.