Imbedding of algebras in algebras of triangular matrices

A. Z. Anan'in

Sbornik. Mathematics, Tome 36 (1980) no. 2, pp. 155-172

Résumé

It is proved in the paper that an algebra $R$ which satisfies identities of the form \begin{gather*} [x,y][z,t][x_1,\dots,x_k]=0,\qquad[[x,y],z][x_1,\dots,x_k]=0,\\ [x_1,y_1]\cdot\dotso\cdot[x_l,y_l]=0, \end{gather*} is imbeddable in the algebra $T_n(K)$ of triangular matrices over a commutative algebra $K$. This permits us to answer both the question due to L. Small concerning the imbeddability of an arbitrary nilpotent algebra in a matrix algebra over a commutative algebra and the question of D. Passman on the imbeddability of a group algebra which satisfies a nontrivial identity in a matrix algebra over a commutative algebra. Bibliography: 6 titles.