ABSTRACT

This chapter attempts to initiate the study of the endomorphism rings, the type endomorphism rings, the automorphism groups, and the tensor products of almost completely decomposable groups. At this point there are more questions than good theorems. The endomorphism ring of an almost completely decomposable group is, as an additive group, again an extension of a completely decomposable group by a finite group, and one possible course of study is to describe its structure and relate it to the structure of the original group. A multitude of questions arise: Which almost completely decomposable groups can be furnished with a ring structure? Which “almost completely decomposable rings” can be the endomorphism ring of an almost completely decomposable group? When is the endomorphism ring commutative? The type endomorphism ring is a finite ring. Its structure can be reasonably well understood but the details are messy. Which finite rings can be realized as the type endomorphism ring of an almost completely decomposable group? The type endomorphism ring determines largely, though not completely the possible direct decompositions of the group and may be a means to attack questions about direct decompositions. Many of the results of this section are from [MS98].