Local Finiteness of Algebras

Abstract

The paper represents a series of comments to the K. A. Zhevlakov and I. P. Shestakov theorem on the existence of a locally finite in the sense of Shirshov over an ideal of the ground ring radical on the class of algebras that are algebraic over this ideal and belong to some sufficiently good homogeneous variety. It is shown in detail how the given theorem includes Plotkin’s and Kuz’min’s theorems on the existence of a locally finite radical on the classes of algebraic Lie and Mal’tsev algebras. There is adduced its generalization to locally finite extensions of ideally algebraic Lie and alternative algebras.