If $R\subset S$ are rings, then why is saying that $R$ is a summand of $S$ as an $R$-module the same as saying that there is an $R$-module homomorphism $S\to R$ that fixes all elements of $R$?
The first condition means that $S\simeq R\oplus R'$ as $R$-modules for some $R$-module $R'$. The only natural map is $R\oplus R'\to R, (r,r')\mapsto r$. But what confuses me is that $R$ isn't a submodule of $S$ now (though it is isomorphic to the submodule $R\oplus \{0\})$.The above map indeed fixes all elements of $R\oplus \{0\}$. But how to deal with the fact that $R\subset S$ is not true?
Also, how to prove the other direction?