Abelian Groups Cyclic over Their Endomorphism Rings

Abstract

In [7] we studied torsion free abelian groups that are finitely generated as modules over their endomorphism rings. It turns out that there is a surprisingly explicit and complete structure (up to quasi-isomorphism, as usual) theory for such groups. We remarked in [7] that groups that are finitely generated over their endomorphism rings arise in various situations in which, once recognized, they can be used to advantage, and we would like to pursue that thought here. Our interest in these groups first developed in [5] in which the abelian groups that are finitely generated and projective over their endomorphism rings were studied. By means of a suitable Horita equivalence one can reduce this study to the case in which the group is actually cyclic (and projective) over its endomorphism ring. For these E-cyclic groups a certain subgroup plays a critical role and we will need it here as well. Since [5] has not yet appeared, and also for the reader’s convenience, we summarize in Section 1 some of the results and definitions from that paper, particularly those related. to the basic tool mentioned above. We apply some of these ideas to study strongly homogeneous groups in Section 2 obtaining some results of Arnold ([1]) from a different and, we think, quite natural point of view and in a more general context. This brings up a question about certain subrings of algebraic number fields which we treat in Section 3. The point of view developed there suggests a more general and, from some points of view, an important problem which is the topic of Section 4.