Let $M, N$ be $R$-modules and $U\subseteq{R}$ a multiplicatively closed subset.
Then $U^{-1}(M\oplus{N})\cong{U^{-1}(M)\oplus{U^{-1}(N)}}$.
Attempt of proof: Let's consider the following exact sequence $$0\longrightarrow{M}\longrightarrow{M\oplus{N}}\longrightarrow{N}\longrightarrow{0}$$ and its localisation $$0\longrightarrow{U^{-1}(M)}\longrightarrow{U^{-1}(M\oplus{N})}\longrightarrow{U^{-1}(N)}\longrightarrow{0}$$ which is exact as well.
Can anyone provide a hint in order to obtain the isomorphism from the exact sequence? Thanks.
Hint: One way (perhaps not the most general, though) is using the fact that an exact sequence of modules $$0\longrightarrow M\stackrel{f}{\longrightarrow} P\stackrel{g}{\longrightarrow} N\longrightarrow 0$$ splits if and only if there exists a map $g':N\to P$ such that $g\circ g'=\operatorname{id}_N$.