Limit T-spaces

Abstract

Let F be a field of prime characteristic p and let V p be the variety of associative algebras over F without unity defined by the identities [[x, y], z] = 0 and x 4 = 0 if p = 2 and by the identities [[x, y], z] = 0 and x p = 0 if p > 2 (here [x, y] = xy − yx). Let A/V p be the free algebra of countable rank of the variety V p and let S be the T-space in A/V p generated by x 21 x 22 ⋯ x 2k + V 2, where k ∈ ℕ if p = 2, and by {ie4170-01}, where k ∈ ℕ and α 1, …, α 2k ∈ {0, p − 1} if p > 2. As is known, S is not finitely generated as a T-space. In the present paper, we prove that S is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative F-algebra without unity of countable rank.