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

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 \(M_{n,n}(E)\) $M_{n,n}(E)$ , where E represents the Grassmann algebra over a field with characteristic \(p > 2\) $p>2$ . 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 \(M_{a,b}(E)\) $M_{a,b}(E)$ , where \(a \ge b\) $a\ge b$ , we establish that it satisfies a standard identity of degree \((a+b)p\) $(a+b)p$ and does not satisfy any standard identity with a degree less than \(2[b(p-1)+a]\) $2[b(p-1)+a]$ .