Notes on linearly compact algebras

Abstract

Let A be a linearly compact ring with ideal neighborhoods of zero, and let N be its radical. Zelinsky shows that [3, Theorem 1 ] A/N is algebraically and topologically isomorphic to a complete direct sum (i.e., a cartesian product) of discrete simple rings with minimum condition. In case A is commutative, then [3, Theorem 2] A is algebraically and topologically isomorphic to a complete direct sum of a radical ring and primary rings with units, all the summands being linearly compact. If A is an algebra and the closure of powers of N has zero intersection, he then shows [4, Theorem C, p. 320] that A (having the usual properties) satisfies the Wedderburn principal theorem. The restriction of N is needed at two stages: raising of orthogonal idempotents of A/N to orthogonal idempotents of A, and the inductive process of producing the semi-simple part. We propose to show that, if A is commutative, the Wedderburn principal theorem is valid without restriction on N. The problem of raising orthogonal idempotents no longer exists, for idempotents which are orthogonal modulo N are already orthogonal; indeed to each idempotent in A/N there is only one idempotent representative in A. By [3, Theorem 2] we can restrict ourselves to primary algebras. Then A/N is a field and we may avail ourselves of the results of field theory to construct the semi-simple part. Our main tool (Lemma 1) is a result of Jacobson [2, Theorem 6]. It also follows easily from this that we can raise a countable number of idempotents with no restriction on the radical. We use the terminology of [3]. The author wishes to acknowledge his thanks to Professor Zelinsky for his many valuable comments.