On Some Identical Relations for Algebras

Abstract

1. In the work of A.I. Malcev [2], results of a general nature are applied in particular to the classification of identical relations of degree 3 for associative algebras. It is shown there that, under natural assumptions on the characteristic, any such identical relation is a linear combination of the following relations:

$$ \sum\limits_{\left( {i_1 ,i_2 ,i_3 } \right)} {x_{i_1 } x_{i_2 } x_{i_3 } = 0,} $$

((1))

$$ \sum\limits_{\left( {i_1 ,i_2 ,i_3 } \right)} {\left( { - 1} \right)^{\sigma _i } x_{i_1 } x_{i_2 } x_{i_3 } = 0,} $$

((2))

$$ x_1 x_2 x_3 + x_2 x_1 x_3 - x_2 x_3 x_1 - x_3 x_2 x_1 = 0, $$

((3))

$$ x_1 x_2 x_3 + x_1 x_3 x_2 - x_3 x_1 x_2 - x_3 x_2 x_1 = 0, $$

((4))

where the summations in relations (1) and (2) are performed over all substitutions, and σ _i is the number of inversions in the permutation (i ₁, i ₂, i ₃) of 1, 2, 3.