Combinatorial generators of the multilinear polynomial identities.
The goal of the work is the study of generators in multilinear elements (i.e., elements that are linear in every indeterminate) of T-ideals of the free associative algebra \(k\langle X\rangle\) with a countable set \(X\) of indeterminates over a field \(k\). The notion of ``generation'' in the set of multilinear elements of a T-ideal is strengthened, and the set of ``combinatorial'' generators is presented analogously to the notion of standard basis for an ideal in the free associative algebra. The problem of existence of a finite system of generators in the set of multilinear elements in a T-ideal is not solved even for characteristic zero. Thus, the author presents a new more ``strict'' version of the well-known Specht problem. Note that in characteristic zero a combinatorial system of generators in the set of multilinear elements of the T-ideal \(T\) generates \(T\) as a total characteristic ideal. The author proves that if a T-ideal contains a multilinear commutator of degree \(3\) or a multilinear product of commutators of degree \(2\), then its set of multilinear elements has a finite set of combinatorial generators. Some multilinear product of commutators of degree \(2\) is a polynomial identity of every finitely generated algebra with nonmatrix polynomial identity, i.e., one that does not hold in the algebra of \((2\times 2)\)-matrices.
scientific article; zbMATH DE number 5358144