A generalization of the Wedderburn-Malcev theorem to infinite dimensional algebras

Abstract

1. Summary. In this paper we extend the Wedderburn Principal Theorem and the Malcev Theorem for associative algebras to certain infinite dimensional algebras. Let A be an algebra over the base field F having (Jacobson) radical N. The Wedderburn Principal Theorem states that if N is nilpotent and if A/N is a finite dimensional separable algebra over F, then A is cleft. The Malcev Theorem, as generalized by Tihomirov [7], states that under these hypotheses any two cleavings of A are conjugate. (See ?2 for the definitions of these terms.) Curtis [3 ] has generalized these theorems to the case where nft IN; = 0 and A is complete with respect to the topology in which the powers of N form a fundamental system of neighborhoods of zero. In ?3 we show that the Wedderburn Principal Theorem holds for such an algebra A in the case where A/N has countable dimension over the base field. To generalize the Malcev Theorem, we drop this dimensionality restriction, but the following hypothesis on the radical is needed: for every positive integer n, Nn/Nn+l is assumed to be complete with respect to a topology in which a fundamental system of neighborhoods of zero is composed of centralizers of finite dimensional separable:subalgebras of A/N (when Nn/Nn+l is considered as an A/N-module). With these conditions it is shown in ?4 that any two cleavings of A are conjugate. The necessity for this additional hypothesis in the Malcev Theorem is shown in ?5 by an example in which A/N is of dimension No over F and N2 =0.