Characteristic ideals and the structure of Lie algebras

Abstract

The purpose of this note is to study the radical and such concepts as semi-simplicity and simplicity for Lie algebras in terms of characteristic ideals. The program follows the outlines of part of a fundamental paper of Fitting on finite groups, and the final result displaying an isomorphism of an algebra V with no solvable characteristic ideals into the Lie algebra of derivations of a uniquely determined completely reducible characteristic ideal in V is an exact analogue of a result of his on semi-simple groups. In case the base field is of characteristic zero, the theory collapses to the classical theory. An example is given to show that this is not the case at prime characteristic.