Adolf Mader

Large E-rings exist

A ring R with 1 is called an E-ring provided Hom,(R, R) z R under the map cp + 1~. The class of E-rings was defined and studied by Schultz [S] in 1973, and further investigated by Bowshell and Schultz [BS] in 1977. Examples of E-rings include Z/(n), subrings of Q, and pure subrings of the ring of p-adic integers. More interesting examples are the torsion-free E-rings of finite rank. These were characterized in [BS] as those rings quasi-isomorphic to R, x R, x . . . x R,, where each Ri is a strongly indecomposable subring of an algebraic number field and Hom,(R,, Rj) = 0 for i #j. In spite of their seemingly specialized nature, such rings have played an important role in the theory of torsion-free groups of finite rank, dating back to Beaumont-Pierce [BPl, BP2], and Pierce [P] in 1960-1961. (See also [APRVW, NR, PV, RI.) Relatively little has been published on infinite rank torsion-free E-rings. Indeed, until recently, the only examples of these were provided by the pure subrings of the p-adic integers. In this paper, the “Black Box” of Shelah is used to construct a host of large E-rings. We show that any torsion-free p-reduced, p-cotorsion-free, commutative ring S may be embed88 0021-8693/87

Almost Completely Decomposable Torsion-Free Abelian Groups

3.00

Regularity and Substructures of Hom

The torsion-free abelian groups of finite rank are the additive subgroups of finite-dimensional vector spaces over the field ℚ of rational numbers. In the following “group” will mean torsion-free abelian group of finite rank unless specifically modified.

Extensions of Abelian Groups

ABSTRACT We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom R (A,M) is regular if and only if Ker(f) ⊆⊕ A and Im(f) ⊆⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom R (A,M). We study regularity in Hom R (A 1 ⊕ A 2, M 1 ⊕ M 2). The existence of a regular function Hom R (A,M) implies the existence of projective summands of Hom R (A,M) End R (A) and of End R ( M ) Hom R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.

Groups and Modules that are Slender as Modules over Their Endomorphism Rings

The object of extension, theory is to give a survey of the possible abstractly different extensions of a group T by a group K. Our solution consists in exhibiting a representative family of extensions of T by K supplemented by a criterion for the isomorphy of groups of this family. We consider arbitrary reduced groups T and arbitrary groups K. The solution is complete in the central case when T is a reduced torsion group and K is torsion-free. For such T and K the results are the following.

Regulating hulls of almost completely decomposable groups

It was shown in Huber [4; Theorem 3.3] that an R-module E is slender as a module over its endomorphism ring S = EndR E if it is of the form E = M(IN) for some R-module M. This had important consequences for questions of E-reflexivity. In Mader [6; Lemma 5.1] it was shown that the abelian group E = M i=1 ∞ Z(pi)is slender over S = End E and this was a key step in proving that the completion of a group A of non-measurable cardinality equipped with the direct-sum-of-cyclics topology of D’Este [1] is algebraically the group HomS(HomZ (A, E), E). At the Honolulu Conference on Abelian Group Theory 1982/83, Martin Huber suggested to me that [6; Lemma 5.1] should be extendable to more general direct sums. In this note we prove such a theorem and use it to characterize completely the totally projective p-groups which are slender as modules over their endomorphism rings.

Representations of Posets and Indecomposable Torsion-Free Abelian Groups

This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

Completions of Linearly Topologized Vectorspaces

Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsion-free abelian groups. For a special class of almost completely decomposable groups we determine the possible ranks of indecomposable groups and show that the possible ranks are realized by indecomposable groups in the class.

A Remak-Krull-Schmidt Class of Torsion-Free Abelian Groups

> 1. In Section III we introduce some particular linear topologies on vectorspaces called functorial topologies. They are first defined as a process which puts a linear topology on every kmvectorspace in such a way that every linear transformation is continuous. These topologies fall into three distinct classes: either (1) every vectorspace is discrete, or (2) every vectorspace is indiscrete, or (3) there exists an infinite cardinal K such that in a vectorspace V a subspace is open if and only if it has codimension <K. In Section IV we determine the completions of vectorspaces with functorial topologies. For the topology of finite codimension the completions are linearly compact spaces, and these are well understood. In all other cases the spaces turn out to be complete provided they have non-measurable dimension. It follows that for any functorial topology an exact sequence of vectorspaces O-+ V, -+ V,+ V3+ 0 implies an exact sequence of completions 0 +

Regularity in Endomorphism Rings

The class of almost completely decomposable groups with a critical typeset of type (1, 5) and a homocyclic regulator quotient of exponent p3 is shown to be of bounded representation type, i.e., in particular, a Remak-Krull-Schmidt class of torsion-free abelian groups. There are precisely 20 near-isomorphism classes of indecomposables all of rank 7, 8, 9.