Jacobson program for Lie algebras

Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a Lie algebra. We give several Lie-module-related characterizations of primitive Lie algebras and prove a representation theorem of a Lie algebra in terms of primitive Lie algebras. We prove a density theorem. Finally, we highlight on the importance of nuclear ideals of prime and semiprime Lie algebras in determining them.