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.
Limit T-subalgebras in free associative algebras
Let F〈X〉 be the free unitary associative algebra over a field F on a free generating set X. An unitary subalgebra R of F〈X〉 is called a T-subalgebra if R is closed under all endomorphisms of F〈X〉. A T-subalgebra R⁎ in F〈X〉 is limit if every larger T-subalgebra W⫌R⁎ is finitely generated (as a T-subalgebra) but R⁎ itself is not. It follows easily from Zorn's lemma that if a T-subalgebra R is not finitely generated then it is contained in some limit T-subalgebra R⁎. In this sense limit T-subalgebras form a “border” between those T-subalgebras which are finitely generated and those which are not. In the present article we give the first example of a limit T-subalgebra in F〈X〉, where F is an infinite field of characteristic p>2 and |X|≥4. Note that, by Shchigolev's result, over a field F of characteristic 0 every T-subalgebra in F〈X〉 is finitely generated; hence, over such a field limit T-subalgebras in F〈X〉 do not exist.
Counterexamples to the Specht problem
The present paper is devoted to the construction of infinitely based varieties of associative algebras over an infinite field of arbitrary positive characteristic.