Eulerian Polynomial Identities and Algebras Satisfying a Standard Identity

Abstract

Using the trivial observation that one can get polynomial identities on R from the ones of Mk(R) we derive from the Amitsur-Levitzki theorem a subset of the identities on n × n matrices, obtained recently by Szigeti, Tuza, and Révész starting from directed Eulerian graphs, which generate the same T-ideal of the free algebra. After that we show that by this method we get a generating set of the T-ideal of identities on the 2 × 2 matrix ring over a field of characteristic 0. We reformulate a problem on algebras satisfying a standard identity in terms of Eulerian identities and use this equivalence in both directions. We apply a result of Braun to Eulerian identities on 3 × 3 matrices, and we give a simpler example which answers the question investigated by Braun. Finally we give an upper estimation on the minimal degree of the standard identity which is satisfied by the matrix algebra over an algebra satisfying some standard identity.