E(R)-algebras that are sharply transitive modules

Abstract

We will consider modules over principal ideal domains that are neither fields nor complete discrete valuation rings and show their wild behavior in two instances, which we explain for simplicity in the category of abelian groups. In the first case we consider constructions of large E-rings: This question was answered in 1987 by Dugas, Mader, Vinsonhaler [M. Dugas, A. Mader, C. Vinsonhaler, Large E-rings exist, J. Algebra 108 (1) (1987) 88-101]. Recall that a ring A is an E-ring if the canonical map End Z A → A (ϕ → ϕ (1)) is an isomorphism. The second question was raised in 1997 by Emmanuel Dror Farjoun and answered recently in [R. Göbel, S. Shelah, Uniquely transitive torsion-free abelian groups, in: Rings, Modules, Algebras, and Abelian Groups, in: Pure Appl. Math., vol. 236, Marcel Dekker, 2004, pp. 271-290]. There is a (non-trivial) torsion-free abelian group G (other than Z and with pure elements) such that its automorphism group Aut G acts sharply transitive on the pure elements of G. These abelian groups G are called UT-groups (or UT-modules for more general rings). Recall that pure elements of a torsion-free abelian group are those divisible only by ±1 and note that automorphisms leave the family pG of pure elements of G invariant. The two apparently distinct questions, can be unified and strengthened. The link between them is the notion of a pure-invertible ring A, which simply means that all pure elements of Z A (seen as abelian group) are units of the commutative ring A. For technical reasons (to have non-trivial pure elements) we also require that A is ℵ 1 -free as abelian group. In the first part of this paper we transform Farjoun's problem into a ring theoretic question on A. It turns out that A must be a pure-invertible ring and an E-ring at the same time. In the second part of the paper we construct such rings assuming ZFC + CH (the special continuum hypothesis). Thus the existence of UT-groups follows and as a byproduct we obtain a new class of E-rings which are at the same time principal ideal domains.