Linear ultrafilters

Abstract

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are “continuous” with respect to U modulo the ideal of those that are “trivial” with respect to U forms a division ring E(U). (These division rings can also be described as the endomorphism rings of the simple left End(X)-modules.) We study this and the dual construction, based on maximal cofiIters of subspaces of X, in particular, the relation between the constructed division rings and the original field or division ring k. We end by examining a more general construction in which X is a module over a general ring, given with both a filter and a cofilter of submodules.