Prescribing endomorphism algebras of ℵ_n-free modules

It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors consider almost-free abelian groups and assume the existence of large canonical, free subgroups. Nevertheless, there exist ℵ₁-separable torsion-free groups G of size ℵ₁ with a basic subgroup B of rank ℵ₁ such that all subgroups of G disjoint from B are also free, but the groups G are still not free. What else can we say about G? The other paper deals with Kaplansky's test problems (which are excellent indicators that the objects defy classification). The authors are able to construct very free abelian groups and verify the test problems for them by a careful choice of particular elements of their endomorphism rings. Accordingly, we want to investigate and construct ℵ_n-free R-modules M (with n an arbitrary, but fixed natural number) over a domain R with End_R M = R for the first time more systematically and uniformly. Recall that M is ℵ_n-free if every subset of size < ℵ_n is contained in a pure, free submodule of M. The requirement End_R M = R implies that M is indecomposable, hence complicated. (We will also allow that End_R M is a prescribed R-algebra, as in the title of this paper.) By now it is folklore to construct such modules M using additional set-theoretic axioms, most notably Jensen's ⋄-principle. In this case the freeness condition can even be strengthened (see [6] and many examples in [9]). However, if we insist on proving this result in ordinary ZFC, then the known arguments fail: The classical constructions from the fundamental paper by Corner [2] do not apply because they are based on pure submodules of p -adic completions of free A-modules, which are never even ℵ₁-free. If we use Shelah's Black Box instead of Jensen's ⋄-principle, then the constructed modules M are still ℵ₁-free, but always fail to be even ℵ₂-free (see [4]). Thus we must develop new methods, which are presented for the first time in Sections 2 to 6, to achieve the desired result (Main Theorem 7.6). With these methods we provide a useful tool for a wide range of problems concerning ℵ_n-free structures which can then be attacked.