Co-Hopfian Abelian groups

E. V. Kaigorodov ; S. M. Chedushev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2015), pp. 21-33

Résumé

In recent years, the interest in co-Hopfian algebraic systems has been growing steadily, with a great number of publications on the topic. However, the studies on co-Hopfian Abelian groups are represented only by individual works. It is therefore natural that there is quite a lot of interesting and important but still open questions related to co-Hopfian Abelian groups. One of these concerns the description of co-Hopfian groups in specific classes of Abelian groups. Consequently, the study of co-Hopfian Abelian groups and their properties is of particular interest. The first section of this paper contains a detailed review of known results on co-Hopfian algebraic systems, the primary emphasis being on co-Hopfian Abelian groups. Special attention is paid to co-Hopfian rings and modules. Some of the major results obtained by specialists in the last half-century are considered in detail. In the second section we obtain the general properties of co-Hopfian Abelian groups. For instance, we prove the co-Hopficity of direct summands of a co-Hopfian Abelian group. We point to one of the cases in which the co-Hopficity of an Abelian group should follow from the co-Hopficity of direct summands in the decomposition of this group. Finally, we give a necessary and sufficient condition of the co-Hopficity of a direct sum of an arbitrary number of Abelian groups on one assumption.

Détail

Keywords: Abelian group, co-Hopfian group, direct sum, fully invariant subgroup, generalized matrix ring.