Co-Hopfian abelian groups | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2015. № 4(36).

Co-Hopfian abelian groups

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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Keywords

абелева группа, кохопфова группа, прямая сумма, вполне инвариантная подгруппа, кольцо обобщенных матриц, Abelian group, co-Hopfian group, direct sum, fully invariant subgroup, generalized matrix ring

Authors

Name	Organization	E-mail
Kaigorodov Evgeniy V.	Gorno-Altaisk State University	gazetaintegral@gmail.com
Chedushev Sergei M.	Gorno-Altaisk State University	S.chedushev@yandex.ru

Всего: 2

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