scholarly journals Modules with the quasi-summand intersection property

1991 ◽  
Vol 44 (2) ◽  
pp. 189-201 ◽  
Author(s):  
Ulrich Albrecht ◽  
Jutta Hausen
Keyword(s):  
Abelian Group ◽  
Endomorphism Ring ◽  
Free Abelian Group ◽  
Intersection Property ◽  
Endomorphism Rings ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Free Modules ◽  
Torsion Free ◽  
Complete Characterisation

Given a torsion-free abelian group G, a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n. If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation cannot be given via endomorphism rings alone but must involve the way in which the endomorphism ring acts on the underlying group. For torsion-free groups G of finite rank without proper fully invariant quasi-summands however, the structure of its quasi-endomorphism ring QE(G) suffices: G has the quasi-summand intersection property if and only if the ring QE(G) is simple or else G is strongly indecomposable.

Commutative Endomorphism Rings

1971 ◽  
Vol 23 (1) ◽  
pp. 69-76 ◽  
Author(s):  
J. Zelmanowitz
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Prime Rings ◽  
Infinite Cyclic Group ◽  
Free Abelian Groups ◽  
Scalar Action ◽  
Torsion Free

The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.

scholarly journals Type radicals and quasi-decompositions of torsion-free abelian groups

1992 ◽  
Vol 52 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Oteo Mutzbauer
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractA composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

Torsion-free abelian groups with optimal Scott families

2018 ◽  
Vol 18 (01) ◽  
pp. 1850002
Author(s):  
Alexander G. Melnikov
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Symbolic Logic ◽  
Technical Result ◽  
Torsion Free Abelian Group ◽  
Scott Family ◽  
Free Abelian Groups ◽  
Torsion Free ◽  
Labeled Tree

We prove that for any computable successor ordinal of the form [Formula: see text] [Formula: see text] limit and [Formula: see text] there exists computable torsion-free abelian group [Formula: see text]TFAG[Formula: see text] that is relatively [Formula: see text] -categorical and not [Formula: see text] -categorical. Equivalently, for any such [Formula: see text] there exists a computable TFAG whose initial segments are uniformly described by [Formula: see text] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [Formula: see text]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) [Formula: see text]-categorical TFAGs for arbitrarily large [Formula: see text] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [Formula: see text]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.

scholarly journals Hypertypes of torsion-free abelian groups of finite rank

1989 ◽  
Vol 39 (1) ◽  
pp. 21-24 ◽  
Author(s):  
H.P. Goeters ◽  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

Let G be a torsion-free abelian group of finite rank n and let F be a full free subgroup of G. Then G/F is isomorphic to T1 ⊕ … ⊕ Tn, where T1 ⊆ T2 ⊆ … ⊆ Tn ⊆ ℚ/ℤ. It is well known that type T1 = inner type G and type Tn = outer type G. In this note we give two characterisations of type Ti for 1 < i < n.

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

2011 ◽  
Vol 21 (08) ◽  
pp. 1463-1472 ◽  
Author(s):  
GÁBOR BRAUN ◽  
LUTZ STRÜNGMANN
Keyword(s):  
Abelian Group ◽  
Finite Automaton ◽  
Free Abelian Group ◽  
Finite Automata ◽  
Additive Group ◽  
Rational Numbers ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

In [Todor Tsankov, The additive group of the rationals does not have an automatic presentation, May 2009, arXiv:0905.1505v1], it was shown that the group of rational numbers is not FA-presentable, i.e. it does not admit a presentation by a finite automaton. More generally, any torsion-free abelian group that is divisible by infinitely many primes is not of this kind. In this article we extend the result from [13] and prove that any torsion-free FA-presentable abelian group G is an extension of a finite rank free group by a finite direct sum of Prüfer groups ℤ(p∞).

scholarly journals Jump degrees of torsion-free abelian groups

2012 ◽  
Vol 77 (4) ◽  
pp. 1067-1100 ◽  
Author(s):  
Brooke M. Andersen ◽  
Asher M. Kach ◽  
Alexander G. Melnikov ◽  
Reed Solomon
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe show, for each computable ordinal α and degree a > 0(α), the existence of a torsion-free abelian group with proper αth jump degree a.

Dynamical entropy for Bogoliubov actions of torsion-free abelian groups on the CAR-algebra

2000 ◽  
Vol 20 (4) ◽  
pp. 1111-1125 ◽  
Author(s):  
VALENTIN YA. GOLODETS ◽  
SERGEY V. NESHVEYEV
Keyword(s):  
Abelian Group ◽  
Unitary Representation ◽  
Free Abelian Group ◽  
Singular Part ◽  
Torsion Free Group ◽  
Torsion Free Abelian Group ◽  
Car Algebra ◽  
Dynamical Entropy ◽  
Free Abelian Groups ◽  
Torsion Free

We compute dynamical entropy in Connes, Narnhofer and Thirring sense for a Bogoliubov action of a torsion-free abelian group $G$ on the CAR-algebra. A formula analogous to that found by Størmer and Voiculescu in the case $G={\Bbb Z}$ is obtained. The singular part of a unitary representation of $G$ is shown to give zero contribution to the entropy. A proof of these results requires new arguments since a torsion-free group may have no finite index proper subgroups. Our approach allows us to overcome these difficulties, it differs from that of Størmer–Voiculescu.

scholarly journals A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

2018 ◽  
Vol 32 (1) ◽  
pp. 319-331
Author(s):  
Mateusz Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Free Abelian Group ◽  
Characteristic Subgroup ◽  
Torsion Free Abelian Group ◽  
Group A ◽  
Free Abelian Groups ◽  
Associative Rings ◽  
Torsion Free ◽  
Group Modulo

Abstract A hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.

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