Abelian Groups pp 613-653 | Cite as

# Endomorphism Rings

## Abstract

With an abelian group *A* one associates the ring \( \mathop{\mathrm{End}}\nolimits A \) of its endomorphisms. This is an associative ring with 1 which frequently reflects several relevant features of the group. Information about direct decompositions is certainly stored in this ring. It is quite challenging to unveil hidden relations between a group and its endomorphism ring.

Quite a lot of information is available for the endomorphism rings of *p*-groups. A celebrated theorem by Baer and Kaplansky shows that torsion groups with isomorphic endomorphism rings ought to be isomorphic. Moreover, the endomorphism rings of separable *p*-groups can be characterized ring-theoretically. However, in the torsion-free case, we can offer nothing anywhere near as informative or complete as for torsion groups. As a matter of fact, on one hand, there exists a large variety of non-isomorphic torsion-free groups (even of finite rank) with isomorphic endomorphism rings, and on the other hand, endomorphism rings of torsion-free groups seem to be quite general: every countable rank reduced torsion-free ring with identity appears as an endomorphism ring of a torsion-free group. The situation is not much better even if we involve the finite topology. The mixed case is of course more difficult to survey.

Though the problem concerning the relations between group and ring properties has attracted much attention, our current knowledge is still far from being satisfactory. The main obstacle to developing a feasible in-depth theory is probably the lack of correspondence between relevant group and relevant ring properties. However, there is a great variety of examples of groups with interesting endomorphism rings, and we will list a few which we think are more interesting. In some cases we have to be satisfied with just stating the results in order to avoid tiresome ring-theoretical arguments. In some proofs, however, we had no choice but to refer to results on rings which can be found in most textbooks on graduate algebra. There are very good surveys on endomorphism rings by Russian algebraists, e.g. Krylov–Tuganbaev [1], and especially, the book by Krylov–Mikhalev–Tuganbaev [KMT].

In this chapter, we require some, but no more than a reasonable acquaintance with standard facts on associative rings.