Examples of associative algebras for which the T-space of central polynomials is not finitely based
In 1988 (see [7]), S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M2(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 k0〈X〉 is finitely based, and it follows from this that every T-space of k1〈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(G0) nor CP(G) is finitely based, thus providing a negative answer to Okhitin’s question.
The Identities and the Central Polynomials of the Infinite Dimensional Unitary Grassmann Algebra Over a Finite Field
We describe the T-ideal of identities and the T-space of central polynomials for the infinite dimensional unitary Grassmann algebra over a finite field.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.