Manfred Dugas

Infinite rank Butler groups

A torsion-free abelian group G is said to be a Butler group if Bext(C, T) = 0 for all torsion groups T. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group G satisfies the T.E.P. over a pure subgroup H if and only if H is decent in G in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called B2-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ^-group. We show under ( V = L) that this is indeed the case for Butler groups of rank Nt. On the other hand it is shown that, under ZFC, it is undecidable whether a group B for which Bext( B, T) = 0 for all countable torsion groups T is indeed a B2-group.

Butler groups of infinite rank. II

A torsion-free abelian group G is called a Butler group if Bext(G, T) = 0 for any torsion group T. We show that every Butler group G of cardinality ltl is a B2-group; i.e., G is a union of a smooth ascending chain of pure subgroups G0 where G+ = Ga + B0, B,0 a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding N., is a B2-group. Moreover, we are able to prove that the derived functor Bext2 (A, T) = 0 for any torsion group T and any torsion-free A with JAI < ,,. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality < Nt, are B2-groups.

Automorphism groups of fields

We consider pairs (K,G) of an infinite field K or a formally real field K and a group G and want to find extension fields F of K with automorphism group G. If K is formally real then we also want F to be formally real and G must be right orderable. Besides showing the existence of the desired extension fields F, we are mainly interested in the question about the smallest possible size of such fields. From some combinatorial tools, like Shelah’s Black Box, we inherit jumps in cardinalities of K and F respectively. For this reason we apply different methods in constructing fields F: We use a recent theorem on realizations of group rings as endomorphism rings in the category of free modules with distinguished submodules. Fortunately this theorem remains valid without cardinal jumps. In our main result (Theorem 1) we will show that for a large class of fields the desired result holds for extension fields of equal cardinality.

Block rigid almost completely decomposable groups and lattices over multiple pullback rings

Abstract Category equivalences from subgroups of finite index in finite direct sums of torsion-free abelian groups of rank 1 to finitely generated lattices over special multiple pullback rings are given. Various examples demonstrate the relevance of representations of finite partially ordered sets over factor rings of the integers to both subjects.

Countable mixed Abelian groups with very nice full-rank subgroups

We exhibit a maximal set of 2N0 “almost rigid” countable mixed abelian groupsG with the same prescribed torsion subgroupstG, the same quotientG/tG and a fixed countable and cotorsion-free ringA such that EndG/Hom(G, tG)≌A. Despite the fact that these candidates will not allow any structure theorem, they are close relatives of the well-behaving family of Warfield groups. The results are developed in a module category for suitable ground rings.

Free modules with two distinguished submodules

Over a commutative ring R with identity, free modules M with 2 distinguished submodules are studied. The category Rep2R of such objects M have the obvious morphisms between them, which are homomorphisms between .R-modules preserving the distinguished submodules. The endo-morphisms for each M constitute a subalgebra EndRM of the algebra EndRM and the readability of λ-generated R-algebras A as EndRM is considered for cardinals λ. Despite the fact that 4 is the minimal number of distinguished submodules for realizing any algebra over a field il, we are able to prove a similar result in Rep2R for many rings R including R = Z and algebras which are cotorsion-free. Several examples illustrate the boarder line of our main result. The main theorem is applied for constructing Butler groups in [11]

Arbitrary torsion classes and almost free abelian groups

Using the set theoretical principle ∇ for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.

Endomorphism algebras of torsion modules II

This paper is part II of [6]. Let R be a fixed integral domain which has a fixed element O ≠ P ∈ R such that \( \mathop \cap \limits_{n \in \omega } {p^n}R = 0. \) If pR is not a maximal ideal and J is an ideal such that pR ⊑ J ≠ R of the form J = annR (x + pnR) for some x ∈ R, then we require |R/J| ≥ 4. In this case we say that R is p-representable. There are many p-representable rings, in particular all archimedian valuation domains the ring of the integers ℤ and the ring Jp of p-adic integers. Then R induces on a R-module A the p-adic topology which is generated by pn A, n ∈ ω. Similarly we say that a R-algebra A is p-representable if R is p-representable, and A is as an R-module the completion of a free R-module in this p-adic topology.

Quasi-E-locally cyclic torsion-free Abelian groups

For a torsion-free abelian group A, we investigate the problem of determining when End(A) is maximal as a ring in the near-ring of all 0-preserving functions on A. We introduce the concept of quasi-End(A)-locally cyclic groups and determine several properties of these abelian groups.

Classification of Modules with Two Distinguished Pure Submodules and Bounded Quotients

In this paper, we consider modules over principal ideal domains R. The objects are free R- modules F with two distinguished pure submodules F0 and F1 with F0 ∩ F1 = 0 and bounded quotient F/(F0⊕ F1) and morphisms are the usual R-homomorphisms which preserve the distinguished submodules. This category is denoted by cRep2.R and its objects, we say the cR2-modules are denoted by F = (F, F0, F1). The rank of a cR2-module F is the rank of the free R-module F. We will show that cR2-@#@ modules are direct sums of indecomposable cR2-modules of rank 1 or 2. The infinite series of indecomposable cR2-modules is well-known and given explicitly after our Main Theorem 1.4. The result was first shown for cR2modules of finite rank in Arnold and Dugas [4], then for countable rank, using heavy machinery due to Hill and Megibben [25] in Files and Göbel [20]. Our proof for arbitrary rank is based on [20] and illustrates the importance of Hill’s notion of an axiom-3 family of modules. The Main Theorem is applied to a classification of Butler groups with two critical types. 1 2