I am trying to read Tsit-Yuen Lam's "Lectures on Modules and Rings" in order to understand why any abelian group can be injected in an injective abelian group, but I got confused by the following proposition on page 61:
After proving this result, he states that a direct sum of injective modules doesn't have to result in an injective module, and that it is only guaranteed in a finite direct sum. Why doesn't this contradict the proposition?