Eulerian polynomial identities on matrix rings (Q1320251)

It is well known that the Amitsur-Levitzki theorem can be proved using some graph-theoretical constructions, see, e.g. \textit{R. G. Swan} [Proc. Am. Math. Soc. 14, 367-373 (1963; Zbl 0118.018)]. The paper under review is a further development in this direction. Assume \(G\) is an Eulerian directed graph having \(k\) vertices and \(N\) edges, and let \(P(G)\) be the set of all covering directed paths of \(G\). Then the ring \(M_ n(R)\) of the \(n\) by \(n\) matrices over an associative commutative unital ring \(R\) satisfies the polynomial (multilinear) identity \(\sum \text{sgn}(p) x_{p(1)} \dots x_{p(N)} = 0\), \(p \in P(G)\) for \(N \geq 2kn\). It is shown that the standard and the double Capelli identities can be obtained in a similar fashion using rather simple Eulerian graphs.