On associative algebras satisfying the Engel condition
\textit{A. I. Kostrikin} [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 3-34 (1959; Zbl 0090.245)] proved that an n-Engel Lie algebra of characteristic p is locally nilpotent if \(p>n\) or \(p=0\). \textit{E. Zel'manov} [Sib. Mat. Zh. 29, No.5, 112-117 (1988; Zbl 0673.17006)] proved that for \(p=0\) such an algebra is even nilpotent, while for \(p\neq 0\) \textit{Yu. Razmyslov}'s [Algebra Logika 10, 33-44 (1971; Zbl 0253.17005)] examples show that it may be nonsolvable. Recently Zel'manov proved that such an algebra is always locally nilpotent (with no restriction on p and n). Here the author proves that an n-Engel Lie algebra \(A^{(-)}\) arising from a finitely generated associative algebra A is Lie-nilpotent. This result is not covered by Zel'manov's one as \(A^{(-)}\) need not be finitely generated. It is also proved that the group of units U(A) is contained in some locally nilpotent variety. The proof is based on some results on PI- algebras such as \textit{A. Braun}'s [J. Algebra 89, 375-396 (1984; Zbl 0538.16013)] on the nilpotency of the radical and Shirshov's solution of Kurosh's problem.
scientific article; zbMATH DE number 4148352