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 tran-sitive. 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. 1