The Existence of Large E(R)-Algebras That Are Sharply Transitive Modules

Abstract

Let R be any principal ideal domain of cardinality < 2ℵ0 , but not a field. We will construct arbitrarily large E(R)-algebras A (of size ≥ 2ℵ0 ) which are at the same time principal ideal domains over R. It follows that the automorphism group of the R-module R A acts sharply transitive on the pure elements of R A. In answering a question of Emmanuel Dror Farjoun, the existence of such large uniquely transitive (UT-modules) for R = ℤ was shown in Göbel and Shelah (2004). The new method, passing first through ring theory, simplifies the arguments; this idea, using localizations, comes from Herden (2005). We applied it recently in Göbel and Herden (2007b) to find such E(R)-algebras of size ≤ 2ℵ0 .