Examples of associative algebras for which the T-space of central polynomials is not finitely based

TL;DR

This paper investigates the finite basis property of T-spaces of central polynomials in associative algebras, showing that for fields of characteristic p > 2, these T-spaces are not finitely based, contrasting previous results.

Contribution

It demonstrates that for prime characteristic p > 2, the T-space of central polynomials in infinite dimensional free associative algebras is not finitely based, providing a counterexample to prior conjectures.

Findings

For characteristic p > 2, T-spaces are not finitely based.

In characteristic zero and 2, T-spaces of central polynomials are finitely based.

Counterexamples to Okhitin's question are constructed for p > 2.

Abstract

In 1988, S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M_2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A with nonzero T-ideal of identities that is properly contained CP(A). V. V. Shchigolev (2001) showed that for any field k of characteristic zero, every T-space of the infinite dimensional free associative k algebra is finitely based, and it follows from this that every T-space of the infinite dimensional free unitary k algebra is also finitely based. This more than answers Okhitin's question (in the affirmative) for fields of characteristic zero. For a field of characteristic 2, the infinite-dimensional Grassmann algebras, unitary and nonunitary, are commutative and thus the T-space of central polynomials of each is finitely based. We shall show in the following that if p is a prime greater than 2 and k is an arbitrary field of characteristic p, then the T-space of central polynomials of the infinite dimension free (unitary or otherwise) associative algebra is finitely based, thus providing a negative answer to Okhitin's question.