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 Central Polynomials of the Finite Dimensional Unitary Grassmann Algebras Over a Finite Field
We describe the T-ideal of identities and the T-space of central polynomials for the unitary finite dimensional Grassmann algebra over a finite field.
On the Identities of the Matrices over the Grassmann Algebra
Abstract The aim of this paper is to find an (upper) bound for the multiplicities of the cocharacters for the identities of the algebra E m -the matrix algebra of order m with entries from the infinite dimensional Grassmann algebra E . We study, for this purpose, in more detail the relation between the ordinary and the graded identities of a given Z 2 -graded algebra that satisfies some specific graded identities (we call them Capelli and anti-Capelli). It turns out that for such algebras there is a pretty close link between the ordinary cocharacters and multiplicities from one hand and the graded ones from the other. The information obtained in Sections V and VI is especially useful for the cocharacters and multiplicities related to "large" diagrams. This examination makes it possible to get an upper bound for the multiplicities of the cocharacters for the identities of E m , that is very similar to the bound for the multiplicities in the case of the identities of the "ordinary" matrices over afield of characteristic zero.