Torsion-free Groups Of Finite Rank Research Articles

Self-small mixed abelian groups G with G/T(G) finite rank divisible

We investigate the class G of self-small mixed abelian groups G with G/T(G) finite rank divisible and the category QG with objects groups in G and maps quasi-homomorphisms. We employ several existing dualities and equivalences between subcategories of QG and subcategories of the category of locally-free torsion-free finite rank abelian groups and quasi-homomorphisms as well as a new duality in QG and new category equivalences from subcategories of QG to certain rational algebras. To aid in our investigations, we also introduce a new invariant and several notions of type for mixed groups. Our results mostly involve classification of subclasses of G in terms of more well-known objects. In particular, we obtain more or less satisfactory descriptions for rank two groups in G, groups in G analogous to cohesive torsion-free groups and groups in G analogous to Murley groups, underlining the resemblance between the mixed groups in G and locally free torsion-free groups of finite rank. For example, we show that every locally free rank two mixed group G is dual to itself via an appropriate mixed group duality.

Quotient divisible abelian groups

An abelian group G G is called quotient divisible if G G is of finite torsion-free rank and there exists a free subgroup F ⊂ G F\subset G such that G / F G/F is divisible. The class of quotient divisible groups contains the torsion-free finite rank quotient divisible groups introduced by Beaumont and Pierce and essentially contains the class G \mathcal {G} of self-small mixed groups which has recently been investigated by several authors. We construct a duality from the category of quotient divisible groups and quasi-homomorphisms to the category of torsion-free finite rank groups and quasi-homomorphisms. Our duality when restricted to torsion-free quotient divisible groups coincides with the duality of Arnold and when restricted to G \mathcal {G} coincides with the duality previously constructed by the authors.

FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS

Abstract This paper investigates torsion-free abelian groups A which are Q E-flat, i.e. for which Q A is flat as an Q E(A)-module. It is shown that a torsion-free A has this property iff Tor1 (M, A) is torsion for all right E(A)-modules M. Furthermore, a torsion-free group of rank 4 is constructed which is Q E-flat but not quasi-isomorphic to an E-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which are Q E-flat.

A note on Fuchs’ Problem 34

We investigate to what extent an abelian group G G is determined by the homomorphism groups Hom ⁡ ( G , B ) \operatorname {Hom}(G,B) where B B is chosen from a set X \mathcal {X} of abelian groups. In particular, we address Problem 34 in Professor Fuchs’ book which asks if X \mathcal {X} can be chosen in such a way that the homomorphism groups determine G G up to isomorphism. We show that there is a negative answer to this question. On the other hand, there is a set X \mathcal {X} which determines the torsion-free groups of finite rank up to quasi-isomorphism.

E-Finitely generated groups

An abelian group G is said to be E-finitely generated (respectively, E-cyclic) if it is finitely generated (resp., cyclic) as a module over its endomorphism ring E. In this paper we refine a theorem of Reid [5] cUid apply the result to settle a question of some years standing concerning infinitely gen¬erated groups. We also determine the structure of torsion-free groups of finite rank for which Q ⊗z is a progenerator over Q ⊗z E

Homomorphic images of completely decomposable finite rank torsion-free groups

Homomorphic images of completely decomposable finite rank torsion-free groups

FINITE AUTOMORPHISM GROUPS OF TORSION-FREE ABELIAN GROUPS OF FINITE RANK

Abelian torsion-free groups of finite rank with finite automorphism groups are considered as rigid extensions of a system of strongly indecomposable groups Aj, j = 1, ..., k, of finite rank and having finite automorphism groups, by a finite p-group P. Such groups are called (A, p)-groups. The author introduces for (A, P)-groups the concept of (A, P)-type, which represents a choice of k integer matrices. A complete description of (A, P)-groups is given by means of (A, P)-types. Using this description, a series of problems on finite groups of automorphisms of torsion-free abelian groups of finite rank are solved. Furthermore, it is shown that the actual solution of any one of these problems comes down to a question of the consistency of a system of equations of the first degree modulo pt, where pt is the maximal order of elements of P. Bibliography: 11 titles.

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 $3.00