The standard identity in algebra $M_{n,n}(E)$

Abstract

The classic Amitsur-Levitzki theorem states that the algebra of square matrices of order over a field K satisfies the standard identity of degree 2n and does not satisfy any other standard identity of a lower degree. This paper presents a result analogous to the Amitsur-Levitzki theorem but applied to the algebra $M_{n,n}(E)$, where E represents the Grassmann algebra over a field with characteristic $p > 2$. More precisely, we demonstrate that the minimal degree of the standard polynomial that makes it a polynomial identity for this algebra is 2np. Furthermore, for the algebra $M_{a,b}(E)$, where $a \ge b$, we establish that it satisfies a standard identity of degree $(a+b)p$ and does not satisfy any standard identity with a degree less than $2[b(p-1)+a]$.