Journal of Group Theory
Editor-in-Chief: Parker, Christopher W.
Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus
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Constructing sharply transitive R-modules of rank ⩽
In this note we will give an elementary proof of the existence of sharply transitive R-modules M over principal ideal domains R. An R-module is sharply transitive (or a UT-module) if its R-automorphism group acts sharply transitively on the pure elements of M. We will assume that M is torsion-free; thus pure elements are simply those elements divisible only by units of R in M. We provide examples of UT-modules of rank ⩽ , while the existence of UT-modules of rank ⩾ was shown recently in Göbel and Shelah [R. Göbel and S. Shelah. Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Applied Math. 236 (Marcel Dekker, 2004), pp. 271–290.] using the more complicated machinery of prediction principles. The existence of countable abelian UT-groups, which follows from this note, was left open in earlier works. Here we require and exploit the existence of algebraically independent elements over the base ring R. (Thus we will need |R| < .) First we will convert the UT problem on modules (as suggested in Herden [D. Herden. Uniquely transitive R-modules. Ph.D. thesis. University of Duisburg-Essen, Campus Essen (2005).]) into a problem on suitable R-algebras. This reduces its solution to a few simple steps and makes the proofs more transparent, requiring only basic results in module theory.
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