Corrigendum et addendum: the Frattini subalgebra of a Bernstein algebra

In a previous paper it is supposed that if A is a Bernstein algebra, every maximal subalgebra, M, verifies that dim M = dim A − 1. This is not true in general. Therefore Proposition 2 in this paper is not correct. However other results there, where this assertion was used, are correct but their proofs need some modifications now.

The Frattini subalgebra of a Bernstein algebra

Let A be a finite-dimensional Bernstein algebra over a field K with characteristic not 2. Maximal subalgebras of A are studied, and they are determined if A is a genetic algebra. It is also proved that the intersection of all maximal subalgebras of A (the Frattini subalgebra of A) is always an ideal. Finally the structure of Bernstein algebras with Frattini subalgebra equal to zero is described.

On ideally finite Lie algebras which are lower semi-modular

The purpose of this paper is twofold: first to correct the statement of Theorem 1 in [4], and secondly to consider related problems in the class of ideally finite Lie algebras.Throughout, L will denote a Lie algebra over a field K, F(L) will be its Frattini subalgebra and φ(L) its Frattini ideal. We will denote by the class of Lie algebras all of whose maximal subalgebras have codimension 1 in L. The Lie algebra with basis {u–1, u0, u1} and multiplication u–1u0 = u–1, u–1u1 = u0, u0u1 = u1 will be labelled L1(0).