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.

References

For example, the set of generators of the T-ideal of $M_n(F)$ is not known unless $n=2$ in which case it is described in .

— Quivers with Polynomial Identities  (2508.00662 - Irelli et al., 1 Aug 2025) in Section 7 (Acyclic quivers)

For example, if $M_n(F)$ is the algebra of $n\times n$ matrices over $F$, we do not know a set of generators for $Id(M_n(F))$ if $n\geq 3.

— Polynomial identities, central polynomials and cocharacters of $M_2(F)$ with transpose superinvolution  (2609.20458 - Santos et al., 17 Sep 2026) in Section 1, Introduction