THE STRUCTURE OF LEFT FILIAL ALGEBRAS OVER A FIELD

Abstract

The structure of left filial algebras over fields is studied. Roughly speaking, these are algebras in which the relation ”being a left ideal” is transitive. It is shown that semiprime algebras are left filial if and only if they are strongly regular and that prime radical algebras are left filial if and only if they are H-algebras, i.e., algebras in which all subalgebras are ideals. In the general case structure theorems describing left filial algebras are obtained. They make it possible to get a complete classification of finite dimensional left filial algebras over some fields.