Combinatorial generators of the multilinear polynomial identities.
type: Item
instance of: scholarly article
title: Combinatorial generators of the multilinear polynomial identities.
zbMATH Open document ID: 1163.16013
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-10-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 5358144
zbMATH Keywords: free associative algebras
zbMATH Keywords: T-ideals
zbMATH Keywords: combinatorial systems of generators
zbMATH Keywords: multilinear polynomial identities
zbMATH Keywords: Specht problem
zbMATH Keywords: Gröbner-Shirshov bases
author: Victor N. Latyshev
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
full work available at URL: s10958-008-0049-5
OpenAlex ID: W2080580609
cites work: Ordering by Divisibility in Abstract Algebras
cites work: Q3789721
cites work: Identities of associative algebras
cites work: A general version of standard basis and its application to T-ideals.
cites work: Gesetze in Ringen. I
DOI: 10.1007/S10958-008-0049-5
description: scientific article; zbMATH DE number 5358144