Direct Sums and Direct Products

Abstract

The concept of direct sum is of utmost importance for the theory. This is mostly due to two facts: first, if we succeed in decomposing a group into a direct sum, then it can be studied by investigating the summands separately, which are, in numerous cases, simpler to deal with. We shall see that almost all structure theorems in abelian group theory involve, explicitly or implicitly, some direct decomposition. Secondly, new groups can be constructed as direct sums of known or previously constructed groups.

Accordingly, there are two ways of approaching direct sums: an internal and an external way. Both will be discussed here along with their basic features. The external construction leads to the unrestricted direct sum, called direct (or cartesian) product, which will also play a prominent role in our future discussions. We present interesting results reflecting the fundamental differences in the behavior of direct sums and products in the infinite case. Pull-back and push-out diagrams will also be dealt with.

Important concepts are the direct and inverse limits that we shall use on several occasions. The final section of this chapter discusses completions in linear topologies.

A reader who is well versed in group theory can skip much of this chapter.