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.     

Definability of separable torsion-free Abelian groups by endomorphism semigroups

In the class of separable torsion-free Abelian groups, we describe those which are defined by their endomorphism semigroups.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 523–530, September-October, 1995.     

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.     

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.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.