Abstract:
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.
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