Polynomial identities of Euler's graphs and matrix rings. (Q2431872)

Let \(R\) be the ring of the \(n\times n\) matrices over a fixed commutative ring with 1. The classical theorem due to Amitsur and Levitzki states that the polynomial identity of the least degree for \(R\) is the standard polynomial \(s_{2n}\). There are several proofs to this theorem. One was proposed by Swan and relies on properties of finite graphs. The authors of the paper under review develop ideas from the proof of Swan and obtain several identities for \(R\).