In reviewing my algebra class material, I "discovered" a strange phenomenon, that the direct product of injective modules is injective, while if $R$ is not noetherian then the direct sum is not necessarily injective. This made me felt very surprised. In fact, this is only true if $R$ is noetherian.
Because it is trivial to prove that the direct summand of an injective module is injective, one is tempted to prove the direct sum of injective modules is injective by regarding it as the direct summand of the direct product. But this cannot be done. Similarly there are examples like $\displaystyle\prod^{\infty}_{i=1}\mathbb{Z}$ which is the direct product of projective modules, but nevertheless not projective itself because it is not free.
My question is, is there something deeper behind this seemingly bizarre dichotomy? Homological algebra is not my specialization field and my knowledge is quite superficial, so I cannot answer it satisfactory myself other than saying it is a reverse of arrows, etc.
There is a post which provides most of the background.