Generalized polynomial identities and rings which are sums of two subrings
The paper under review is devoted to the problem whether a ring which is a sum of two PI-subrings also satisfies a polynomial identity. The authors solve the problem for an important class of subrings satisfying a nonmatrix identity. The main result is the following. Let \(\nu_m\) be the class of all rings satisfying the polynomial identity \([x_1, y_1] \dots [x_m, y_m] = 0\). If \(A \in \nu_m\), \(B \in \nu_n\), then \(R = A + B\) satisfies a polynomial identity. This theorem generalizes a recent result of \textit{Yu. Bakhturin, A. Giambruno} [Rend. Circ. Mat. Palermo, II. Ser. 43, No. 2, 250-258 (1994; Zbl 0829.16014)] that the sum of two commutative subrings satisfies the identity \([x_1, y_1] [x_2, y_2] = 0\), i.e. \(A, B \in \nu_1\) implies \(R = A + B \in \nu_2\). The proof uses generalized polynomial identities and is based on the techniques developed by \textit{M. Ferrero, E. R. Puczyłowski} [Arch. Math. 53, No. 1, 4-10 (1989; Zbl 0645.16005)]. The authors also establish that if \(A, B \in \nu_2\), then \(R = A + B\) satisfies some power of the standard identity \(s_6(x_1, \dots, x_6) = 0\) and show that the \(3 \times 3\) matrix algebra over a field is a sum of two subalgebras from \(\nu_2\).
scientific article; zbMATH DE number 862290
scientific article