We’ve covered direct sums in the case of abelian groups — that is, -modules — but the concept extends to modules over arbitrary rings. This gives me a good chance to go back and clean up my coverage.
We build the direct sum of two -modules and on the product of the underlying sets. We define an abelian group structure by , and an action of by .
Now, here’s the diagram:
The direct sum of two left -modules and comes equipped with four module homomorphisms. For we have the pair and . These are defined as follows:
There is a similar pair with a similar definition for . These homomorphisms satisfy the identities
where is the identity homomorphism on the module , and is the homomorphism between two modules sending every element of the domain to the element in the codomain.
Now if we have any two homomorphisms and from a module to and respectively, then there is a unique homomorphism making the two triangles on the top commute. That is, , and similarly for . In fact, we can define , for then
and so on. Similarly, given two homomorphisms and from and to a module , then the homomorphism is the unique homomorphism making the lower two triangles commute.
The upshot of all this is that the direct sum behaves like both the direct product and the free product of two groups, since it satisfies both universal properties. For any finite number of modules we can build the direct sum and it also satisfies the analogous universal properties, and comes equipped with analogous injections and projections satisfying analogous relations to those above.
The upshot of all this is that the direct sum of a finite collection of -modules behaves like both the direct product and the free product of groups. In fact, we can take that as the definition, derive the relations between the injections and projections, and use the above construction to show such a thing actually exists. On the other hand, we can take the relations between injections and projections as the definitions, use the construction to show existence, and derive the universal property from the relations as above.
For an infinite index set the situation is a bit more complicated. Here we use the definition from the injections and projections with the specified relations. Then the underlying set of the infinite direct sum is not the infinite cartesian product of the underlying sets. It’s actually the list of all such “-tuples” where all but a finite number of the entries are the elements of the respective modules. This only satisfies the universal property for the — the bottom of the diagram above. For the top we really do need the infinite direct product of the modules, which uses the whole infinite cartesian product of the underlying sets. However, for most purposes the direct sum is all we need, and the relations between the injections and the projections are the most useful part of this definition.