Hi I am refering to proposition 2.14 in Atiyah-MacDonald introduction to commutative algebra and I can't find the bilinear maps that will induce $A$-module homomorphisms $f,g$ where $f:(M \oplus N) \otimes P \to (M \otimes P) \oplus (N \otimes P)$ , $ g :(M \otimes P) \oplus (N \otimes P) \to (M \oplus N) \otimes P $
and $f \circ g = id$ and $g \circ f = id $, hence $f$ and $g$ are isomorphisms.
Can anyone please help me?
Thanks!
The map $f$ is induced from $$(M \oplus N) \times P \rightarrow (M \otimes P) \oplus (N \otimes P),$$ $$((m,n),p) \mapsto ((m,p),(n,p))$$ while $g$ is obtained by first inducing maps $$g_1 : M \otimes P \rightarrow (M \oplus N) \otimes P,$$ $$g_2 : N \otimes P \rightarrow (M \oplus N) \otimes P$$ from the bilinear maps $$M \times P \rightarrow (M \oplus N) \otimes P,$$ $$(m,p) \mapsto ((m,0),p)$$ and $$N \times P \rightarrow (M \oplus N) \otimes P,$$ $$(n,p) \mapsto ((0,n),p).$$ Then using the universal property of direct sum induces the desired map $g$.
