If $R$ is a ring, is there a simple way of showing that the functor $R^2 \otimes -$ from the category of $R$-modules to the one of abelian groups is left-exact?
I know it is right-exact, I have a suspicion it should also be left exact. Now I know it is not true if we simply replace $R^2$ by $R$ and I think the way I am proceeding would prove that wrong result too... So I know I'm not doing this the right way and I still don't know why...
Now I think I manage to break the proof down to showing that for an $R$-module $N$, $R^2 \otimes N$ is the same as the direct sum of two copies of $N$. I still don't see how to manage that...
I am a bit lost with category theory so any help would be very much appreciated!