David M. Arnold

Finite rank torsion free Abelian groups and rings

Notation and preliminaries.- Types and rank-1 groups.- Examples of indecomposable groups and direct sums.- Endomorphism rings and decompositions of rank 2 groups.- Pure subgroups of completely decomposable groups.- Homogeneous completely decomposable groups and generalizations.- Completely decomposable groups and generalizations.- Additive categories, quasi-isomorphism and near-isomorphism.- Stable range, substitution, cancellation, and exchange properties.- Subrings of finite dimensional Q-algebras.- Orders in finite dimensional simple Q-algebras.- Maximal orders in finite dimensional simple Q-algebras.- Near isomorphism and genus class.- Grothendieck groups.- Additive groups of subrings of finite dimensional Q-algebras.- Q-simple and p-simple groups.

Strongly homogeneous torsion free abelian groups of finite rank

An abelian group is strongly homogeneous if for any two pure rank 1 subgroups there is an automorphism sending one onto the other. Finite rank torsion free strongly homogeneous groups are characterized as the tensor product of certain subrings of algebraic number fields with finite direct sums of isomorphic subgroups of Q, the additive group of rationals. If G is a finite direct sum of finite rank torsion free strongly homogeneous groups, then any two decompositions of G into a direct sum of indecomposable subgroups are equivalent. D. K. Harrison, in an unpublished note, defined a p-special group to be a strongly homogeneous group such that G/pG Z/pZ for some prime p and qG = G for all primes q =# p and characterized these groups as the additive groups of certain valuation rings in algebraic number fields. Richman [7] provided a global version of this result. Call G special if G is strongly homogeneous, G/pG = 0 or Z/pZ for all primes p, and G contains a pure rank 1 subgroup isomorphic to a subring of Q. Special groups are then characterized as additive subgroups of the intersection of certain valuation rings in an algebraic number field (also see Murley [5]). Strongly homogeneous groups of rank 2 are characterized in [2]. All of the above-mentioned characterizations can be derived from the more general (notation and terminology are as in Fuchs [3]): THEOREM 1. Let G be a torsion free abelian group of finite rank. Then G is strongly homogeneous iff G R 0 z H where H is a finite direct sum of isomorphic torsion free abelian groups of rank 1, R is a subring of an algebraic number field K (with IK E R), and every element of R is an integral multiple of a unit in R. PROOF. (e) First of all, the additive group of R, denoted by R +, is strongly homogeneous. Let X and Y be pure rank 1 subgroups of R +. There are units u and v of R in X and Y, respectively. Left multiplication by vu 1 induces an automorphism g of R + with g(X) = Y. Secondly, R 0zA is strongly homogeneous, where A is a torsion free abelian group of rank 1 with H Sk= 1 0 A. Choose 0 7# a e A and define 8: R + R 0z A by 8(r) = r 0 a. Then S is a monomorphism. LetX and Y Received by the editors January 20, 1975. AMS (MOS) subject classifications (1970). Primary 20K15.

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.

Representations of Finite Posets Over Discrete Valuation Rings

Given a discrete valuation ring R, a non-negative integer m, and a finite partially ordered set S, the representation type of a category rep(S*, R, m) of finite rank free R-module representations of S is characterized in terms of S and m. As an application, representation types of some categories of torsion-free abelian groups of finite rank are determined.

A class of pure subgroups of completely decomposable abelian groups

The class M of pure subgroups of completely decomposable torsion free abelian groups of finite rank is considered by Butler [2]. Many of the examples of pathological direct sum decompositions of finite rank torsion free abelian groups occur in the class .4. On the other hand, homogeneous a-groups are completely decomposable and if A is an s-group, then typeset(A) is finite and closed under intersection of types. This note considers a-groups with typesets of cardinality at most 4. Motivation is provided by the curious results of Cruddis [3]. An easier and more illuminating proof of these results is a consequence of the following theorems. Let A= {-To, TI, T2, T73} be a set of types with To =rl1r -72r lT3; let .A be the class of a-groups with typeset c A; let :A be the sum of types in A; and let >,A be the sum of types in typeset(A). Define T to be the type represented by (Xo) and, forp a prime, let T-D be the type represented by (ne) where n2,=O and nQ= xo if p#q.

Isomorphism invariants for abelian groups

Let A=(A 1 ,...,A n ) be an n-tuple of subgroups of the additive group, Q, of rational numbers and let G(A) be the kernel of the summation map A 1 ○+...○+A n →∑A i and G[A] the cokernel of the digonal embedding ∩A i →A 1 ○+...○+A n . A complete set of isomorphism invariants for all strongly indecomposable abelian groups of the form G(A), respectively, G[A], is given. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of Warfield groups

Co-purely indecomposable modules over discrete valuation rings

Abstract A co-purely indecomposable module is a quasi-isomorphic dual of a pure finite rank submodule of the completion of a discrete valuation ring. Co-purely indecomposable modules are classified up to isomorphism.

Representation type of finite rank almost completely decomposable groups

Abstract An almost completely decomposable (acd) group is afiniterank torsion-free abelian group containing a finite direct sum of rank-1 groups as a subgroup of finite index. This paper is devoted to a determination of the representation type of indecomposables in categories of acd groups. In the process, representation types of categories of related representations of finite partially ordered sets (posets) over the rings Z/pjZ and Zp , the localization of the integers Z at a prime p, are also determined.

Direct sums of local torsion-free abelian groups

The category of local torsion-free abelian groups of finite rank is known to have the cancellation and n-th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960s.

Finite Rank Butler Groups

Free groups are the projectives and divisible groups are the injectives for the category of torsion-free abelian groups [Hungerford 74]. Changing the category and the class of defining homomorphisms can change the projectives and injectives. Properties of Coxeter correspondences and almost split sequences are applied to categories of finite rank Butler groups in this section.