Eulerian polynomial identities and algebras satisfying a standard identity
Let \(M_ n(K)\) be the ring of all \(n \times n\)-matrices over a field \(K\). In a paper of \textit{J. Szigeti, Z. Tuza} and \textit{G. Révész} [J. Algebra 161, 90-101 (1993; Zbl 0802.16015)] a new class of polynomial identities of \(M_ n(K)\) was found. These identities are constructed starting from directed connected graphs, and the author calls them Eulerian identities. In section 2 of the paper under review it is shown that the construction of Eulerian identities is a special case of the general method to obtain polynomial identities of a ring \(R\) from identities of \(M_ k(R)\). In Section 3 the author gives a subset of the identities of J. Szigeti, Z. Tuza, and G. Révész which generates the same \(T\)-ideal of the free algebra as all these identies. In Section 4 an identity basis of \(M_ 2(K)\), (where \(K\) has characteristic 0) consisting of two Eulerian identities is found (an identity basis of \(M_ 2(K)\) was found by \textit{Yu. P. Razmyslov} [Algebra Logika 12, 83- 113 (1973; Zbl 0282.17003)] and \textit{V. Drensky} [ibid. 20, 282-290 (1981; Zbl 0496.16017)]). In Section 5 the author reformulates in terms of Eulerian identities the following question investigated by \textit{A. Braun} [Isr. J. Math. 55, 345-349 (1986; Zbl 0615.16010)]: whether \(M_ 2(R)\) satisfies \(s_{4n} = 0\) whenever \(R\) satisfies \(s_{2n} = 0\), where \(s_ k\) is a standard identity of degree \(k\). In the paper mentioned A. Braun shows that the answer is negative: he gives an example of a \(K\)-algebra \(R\) such that \(s_ 6\) holds in \(R\) but \(s_{12}\) fails in \(M_ 2(R)\). In the paper under review a simpler example is given: the author proves that there exists a \(K\)-algebra \(R\) such that \(s_ 4\) holds in \(R\) but \(s_ 8\) fails in \(M_ 2(R)\) (Theorem 5.4). The author proves also that all Eulerian identities of \(M_ 3(K)\) are consequences of \(s_ 6\) (Corollary 5.3). Finally, Theorem 5.5 states that if an algebra \(R\) satisfies \(s_{2n} = 0\) then \(M_ k(R)\) satisfies \(s_ q = 0\) where \(q = 2nk^ 2 - k^ 2 + 1\).
scientific article; zbMATH DE number 701285