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

Abstract

In 1988 (see [7]), 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 for which 0 ≠ T(A) ⊊ CP(A). V. V. Shchigolev (see [9], 2001) showed that for any field of characteristic zero, every T-space of k 0〈X〉 is finitely based, and it follows from this that every T-space of k 1〈X〉 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 > 2 and k is an arbitrary field of characteristic p, then neither CP(G 0) nor CP(G) is finitely based, thus providing a negative answer to Okhitin’s question.