View source for Locally nilpotent algebra

An algebra in which any finitely-generated subalgebra is nilpotent (cf. [[Nilpotent algebra|Nilpotent algebra]]; [[Finitely-generated group|Finitely-generated group]]). It is convenient to represent a locally nilpotent algebra as the union of an increasing chain of nilpotent subalgebras. A locally nilpotent [[Algebra with associative powers|algebra with associative powers]] is a [[Nil algebra|nil algebra]]. A locally nilpotent Lie algebra is an [[Engel algebra|Engel algebra]]. The class of locally nilpotent algebras is closed with respect to homeomorphic images and transition to subalgebras.

In the case of associative algebras an extension of a locally nilpotent algebra by a locally nilpotent algebra is again a locally nilpotent algebra. Therefore, the sum of all locally nilpotent ideals of an associative algebra is the largest locally nilpotent ideal that contains all locally nilpotent ideals; it is called the Levitskii radical. An analogue of the Levitskii radical can be defined in an Engel Lie algebra of bounded index. A locally nilpotent algebra cannot be simple (cf. [[Simple algebra|Simple algebra]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.N. Herstein,   "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,   "Lie rings satisfying the Engel condition"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''21'''  (1957)  pp. 515–540  (In Russian)</TD></TR></table>