Prescribing endomorphism algebras of $\aleph_n$-free modules

Abstract

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 ℵ 1 -separable torsion-free groups G of size ℵ 1 with a basic subgroup B of rank ℵ 1 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 ℵ 1 -free. If we use Shelah's Black Box instead of Jensen's ♦-principle, then the constructed modules M are still ℵ 1 -free, but always fail to be even ℵ 2 -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.

Kaplansky Test Problems for $R$-Modules in ZFC

“…This result extends several known statements starting with Corner [2], and a special case of it, when R is the ring of integers, was obtained recently by Göbel-Herden-Shelah [21,Corollary 9.1(iii)].…”

Separable $\aleph_k$-free modules with almost trivial dual

“…For non-hereditary rings however, it is necessary to modify this notion. The following more general definition of κ-freeness is due to Göbel, Herden, Shelah [6], which is a slightly stronger version of that in Eklof, Mekler [3]. The first step to achieve ℵ k -freeness is to prove a Freeness-Proposition, which allows us to enumerate subsets of Λ in such a convenient way that we can prove linear independence in the constructed modules.…”

“…The Black Box is a combinatorial principle that allows us to partially predict a given map under specific cardinal conditions. Various variants of this principle have been successfully used to realize complicated algebraic constructions (see, for example, [2], [6], [9] and [14] for applications of the General Black Box and the Strong Black Box). Its main feature is the fact that it is provable in ZFC, since prediction of maps is normally the direct consequence of additional set-theoretic assumptions like Martin's Axiom or Jensen's Diamond Principle ♦.…”

“…Only in 2007 Shelah [15] introduced a new, more powerful version of his Black Box principle, which allowed him to construct ℵ n -free abelian groups with trivial dual. This breakthrough immediately led to other constructions of ℵ n -free groups and modules with different algebraic properties like almost trivial dual or prescribed endomorphism ring (see for example [6], [7] and [8]).…”

$\aleph_k$-free cogenerators

“…Some first sporadic examples of non-free ℵ k -free groups for integers k ≥ 2 can be found in [10,13], however, the breakthrough in constructing ℵ k -free groups with prescribed additional properties is more recent. In [6,15], ℵ k -free groups with trivial dual were constructed, and [5] provides a construction for ℵ k -free groups with prescribed endomorphism rings. Similar constructions of ℵ k -free groups and modules for k ≥ 2 can be found in [4,7,11,12] and are based on the λ-Black Box as a guiding combinatorial principle.…”