# The Minimal Degree Standard Identity on $M_nE^2$ and $M_nE^3$

## TL;DR

This paper establishes a new lower bound for the minimal degree of standard identities in matrix rings over Grassmann algebras, using combinatorial methods related to Eulerian trails.

## Contribution

It provides the first sharp bounds for standard identities in matrix rings over Grassmann algebras, extending Amitsur--Levitzki-type theorems with combinatorial proofs.

## Key findings

- Lower bound of 2⌊m/2⌋+4n-4 for standard identities
- Bound is sharp for m=2,3 and all n≥2
- Purely combinatorial proof using Eulerian trails

## Abstract

We prove an Amitsur--Levitzki-type theorem for Grassmann algebras, stating that the minimal degree of a standard identity that is a polynomial identity of the ring of $n \times n$ matrices over the $m$-generated Grassmann algebra is at least $2\left\lfloor\frac{m}{2}\right\rfloor+4n-4$ for all $n,m\geq 2$ and this bound is sharp for $m=2,3$ and any $n\geq 2$. The arguments are purely combinatorial, based on computing sums of signs corresponding to Eulerian trails in directed graphs.