Limit T-subalgebras in free associative algebras

Let F〈X〉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〉F〈X〉 is called a T-subalgebra if R   is closed under all endomorphisms of F〈X〉F〈X〉. A T  -subalgebra R⁎R⁎ in F〈X〉F〈X〉 is limit if every larger T  -subalgebra W⫌R⁎W⫌R⁎ is finitely generated (as a T  -subalgebra) but R⁎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⁎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〉F〈X〉, where F   is an infinite field of characteristic p>2p>2 and |X|≥4|X|≥4. Note that, by Shchigolev's result, over a field F of characteristic 0 every T  -subalgebra in F〈X〉F〈X〉 is finitely generated; hence, over such a field limit T  -subalgebras in F〈X〉F〈X〉 do not exist.