Let $M, N, P$ be $A$-modules, where $A$ is a commutative ring with identity. I want to prove that $(M \oplus N) \otimes P$ is isomorphic to $(M \otimes P) \oplus (N \otimes P)$.
I start by defining a map $(M \oplus N) \times P \rightarrow (M \otimes P) \oplus (N \otimes P)$ by $(x+y,z) \mapsto (x \otimes z, y \otimes z)$. This map is $A$-bilinear, thus it induces a homomorphism $f:(M \oplus N) \otimes P \rightarrow (M \otimes P) \oplus (N \otimes P)$ such that $(x,y) \otimes z \mapsto (x \otimes z, y \otimes z)$.
Now i want to construct a homomorphism the other way, i.e. $g: (M \otimes P) \oplus (N \otimes P) \rightarrow (M \oplus N) \otimes P$ and show that $g \circ f, g \circ g$ are the identity maps. I am having difficulty in constructing a suitable bilinear map that will induce $g$.
Any insights? Thanks.