Definability of Abelian groups by homomorphism groups and endomorphism semigroups

In the paper, the isomorphism problem for completely decomposable Abelian torsion-free groups of finite rank is treated under the assumption that the groups of homomorphisms of these groups into some Abelian group are isomorphic and, moreover, the endomorphism semigroups of the groups are isomorphic.     

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.     

Definability of vector groups by endomorphism semigroups

Some conditions for torsion-free Abelian groups to be isomorphic are found; the groups in question are decomposable into a direct product of rank 1 groups with isomorphic endomorphism semigroups.Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 422–428, July-August, 1994.     

Complete direct sums of Abelian torsion-free groups of rank 1 with isomorphic groups or rings of endomorphisms. I

Isomorphism conditions are established for torsion-free Abelian groups with isomorphic groups or rings of endomorphisms decomposable into complete direct sums of groups of rank 1.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 867–878, December, 1973.     

Determination of the Direct Sums of Rational Groups by H-Representations of the Endomorphism Rings up to Equality

The problem of determination of Abelian groups (up to isomorphism) by their rings of endomorphisms in the class of completely decomposable torsion-free Abelian groups has been solved earlier. For the class of direct sums of rational groups, one can speak of determination of Abelian groups by rational representations of their endomorphism rings up to equality. In this paper, we consider this problem for the class of finite direct sums of rational groups and for some subclasses.     

The question of the definability of Abelian groups by the centers of their endomorphism rings

We obtain necessary and sufficient conditions for an Abelian group in the class of completely decomposable torsion-free Abelian groups of a chosen finite rank to be definable in this class by the center of the endomorphism ring of the group.     

On Torsion-Free Minimal Abelian Groups

ABSTRACT

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. In this article we show that torsion-free groups which are complete in their ?-adic topology or are of p-rank not greater than 1, for all primes p, are minimal. A criterion is found for the minimality of all finite rank and for large classes of infinite rank completely decomposable groups. Separable minimal groups are also considered.     

Cobalanced Extensions of a Torsion-Free Module by a Torsion Module Over an Integral Domain

Cobalanced extensions of torsion-free modules by torsion modules over domains are investigated. As in the case of Abelian Groups, these sequences will split if and only if the torsion-free module is a pure submodule of a vector module. Unlike this case, even the torsion-free modules of finite rank need not be locally completely decomposable. For 1-dimensional Noetherian domains, necessary and sufficient conditions are given for that to be true.     

Normal subgroups of holomorphs of Abelian groups and almost holomorphic isomorphism

Two groups are said to be almost holomorphically isomorphic if each of them is isomorphic to a normal subgroup of the holomorph of the other group. In this paper, we study normal subgroups of holomorphs of Abelian groups and highlight some torsion-free groups for which almost holomorphic isomorphism implies isomorphism. In addition, we consider the problem of determinateness of a group by its holomorph. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 9–16, 2007.     

Torsion-free groups and modules with the involution property

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of p-adic integers.     

Computation of the Mapping of Factorization by the Radical for K 0 + of the Endomorphism Ring

The mapping of factorization by the radical is computed for the semigroup of projective, finitely generated modules over the endomorphism ring of an almost completely decomposable torsion-free Abelian group of finite rank that is divisible by almost all prime numbers. Also, an answer is given to the question concerning the collections of groups of rank 1 for which one can construct an almost completely decomposable group, indecomposable as an object in , by adding a generator. Bibliography: 4 titles.     

Determinability of a Completely Decomposable Block-Rigid Torsion-Free Abelian Group by Its Automorphism Group

In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable block-rigid torsion-free Abelian groups.     

On the definability of completely decomposable torsion-free Abelian groups by endomorphism rings and some groups of homomorphisms

Mathematical Notes - In the present paper,we describe conditions on a vector group C which are necessary and sufficient for the class of completely decomposable torsion-free Abelian groups to be a...     

On fully inert subgroups of completely decomposable groups

Mathematical Notes - The completely decomposable torsion-free Abelian groups with finitely many homogeneous components for which every fully inert subgroup is commensurable with a fully invariant...     

On classification of the Butler and almost completely decomposable groups of finite rank

An approach to the study of torsion-free Abelian groups of finite rank developed by the author in an earlier paper is applied to smaller classes of Butler groups and of almost completely decomposable groups. Bibliography:3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 125–139.     

Quasi-Endomorphism Rings of Strongly Indecomposable Torsion-Free Abelian Groups of Rank 4 With Pseudosocles of Rank 1

We obtain the matrix representation of the minimal rational algebra whose subalgebras, up to isomorphism, are quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 4 with pseudosocles of rank 1.     

The Jacobson Radical of an Endomorphism Ring for an Abelian Group

The Jacobson radical of an endomorphism ring is computed for a completely decomposable torsion-free Abelian group and for a mixed Abelian group in one class of mixed groups. For the latter case, we also look into the question when a factor ring w.r.t. the radical is regular in the sense of Nuemann.     

Transfinite Diameter with Respect to the Neumann Function

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.     

Isomorphism conditions for completely decomposable torsion-free abelian groups with isomorphic rings of endomorphisms

Assuming the generalized continuum hypothesis, we obtain necessary and sufficient conditions for completely decomposable torsion-free abelian groups with isomorphic rings of endomorphisms to be isomorphic.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 403–408, April, 1972.I. H. Bekker has devoted much attention to this work and the author is deeply grateful to him.