Ekaterina Blagoveshchenskaya, St. Petersburg Polytechnical University

Abstract:
An almost completely decomposable group (acd-group) is a torsion-free abelian group of finite rank containing a completely decomposable group (i.e. a direct sum of rank-one groups) as its subgroup of finite index. The theory of acd-groups has been intensively developed in the last decades. Monograph "Almost Completely Decomposable Groups" by A. Mader, [1], collects a wide variety of the results obtained. In general, group properties are tightly connected with its endomorphism ring properties. Another monograph "Endomorphism Rings of Abelian Groups" by P. Krylov, A. Mikhalev, A. Tuganbaev, [2], provides a comprehensive foundation for the investigation of such links since it includes the classical background as well as recent results connecting abelian groups with their endomorphism rings. The main basis for combining group and ring approaches in the particular case of acd-groups is the well-known fact that endomorphism rings End X of acd-groups X are also acd-groups as additive structures. In this way some special methods, which appeared in the acd-group theory, can be applied to the rings for examining dual connections between acd-groups and their endomorphism rings, see [3].

References

[1] A. Mader. Almost Completely Decomposable Abelian Groups, Gordon and Breach, Algebra, Logic and Applications, Vol. 13, Amsterdam, 2000.
[2] P. Krylov, A. Mikhalev and A. Tuganbaev. Endomorphism Rings of Abelian Groups, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003.
[3] E. Blagoveshchenskaya. Dualities between almost completely decomposable groups and their endomorphism rings, Journal of Mathematical Sciences, Kluwer Academic Publishers, V. 131, 5, pp. 5948-5961, 2005.