I am thinking about the proof of the second isomorphism theorem, and something isn't very clear to me.
Let $R$ be a ring ,$S\subset R$ a subring and $I\subset R$ an ideal. We have the natural homomorphism $f:R\rightarrow R/I$. The theorem states that $Im(S)=S+I/I$. My question is why not simply $Im(S)=S/I$?
I understand that it is not true (for starters, $S/I$ need not be a subring), but I cannot explain that to my self in a convincing way.