# Locally nilpotent algebra

An algebra in which any finitely-generated subalgebra is nilpotent (cf. Nilpotent algebra; 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 is a nil algebra. A locally nilpotent Lie algebra is an 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).

#### References

[1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |

[2] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) |

[3] | A.I. Kostrikin, "Lie rings satisfying the Engel condition" Izv. Akad. Nauk SSSR Ser. Mat. , 21 (1957) pp. 515–540 (In Russian) |

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Levitskii radical.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Levitskii_radical&oldid=39416