We describe the T -space of central polynomials for both the unitary and the nonunitary infinite dimensional Grassmann algebra over a field of characteristic p = 2 (infinite field in the case of the unitary algebra).

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.

We study the Grassmann T -space, S 3 , generated by the commutator [x 1 , x 2 , x 3 ] in the free unital associative algebra K x 1 , x 2 , . . . over a field of characteristic zero. We prove that S 3 = S 2 ∩ T 3 , where S 2 is the commutator T -space generated by [x 1 , x 2 ] and T 3 is the Grassmann T -ideal generated by S 3 . We also construct an explicit basis for each vector space S 3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x 1 , . . . , xn, and deduce the recursive vector space decomposition T 3 ∩ Pn = (S 3 ∩ Pn) ⊕ (T 3 ∩ P n−1 )xn.

No abstract

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.