# Dummit and Foote Proposition 10.29 proof

Let $$M,N,P$$ be $$R$$-modules, prove that $$\text{Hom}_R(M \oplus P,L) \cong \text{Hom}_R(M,L)\oplus \text{Hom}_R(P,L)$$.

Attempt:Let $$\pi_1:M \oplus P\rightarrow M$$ and $$\pi_2:M \oplus P \rightarrow P$$ be the natural projections. If $$f \in \text{Hom}_R(M\oplus P,L)$$ then $$f \circ \pi_1$$ and $$f \circ \pi_2$$ determine elements in $$\text{Hom}_R(M,L)$$ and $$\text{Hom}_R(P,L)$$ respectively. Thus $$f:\text{Hom}_R(M \oplus P,L) \rightarrow \text{Hom}_R(M,L)\oplus \text{Hom}_R(P,L)$$ given by $$f \mapsto (f \circ \pi_1) \times (f \circ \pi_2)$$ determines a homomorphism. Let $$f_1 \in \text{Hom}_R(M,L),f_2 \in \text{Hom}_R(P,L)$$, and define $$f \in \text{Hom}_R(M \oplus P,L)$$ by $$f=f_1 \oplus f_2$$. Then we have that $$\text{Hom}_R(M,L)\oplus \text{Hom}_R(P,L) \rightarrow \text{Hom}_R(M \oplus P,L)$$ given by $$f_1 \times f_2 \mapsto f_1 \oplus f_2$$ is the inverse of the first map. Let $$\Phi$$ be the first map, and $$\Psi$$ the second. We have $$$$\begin{split} (\Phi \circ \Psi)(f_1 \times f_2))=&\Phi \circ (f_1 \oplus f_2)\\ =&((f_1 \oplus f_2) \circ \pi_1) \times ((f_1 \oplus f_2) \circ \pi_2))\\ =&f_1 \times f_2 \\ \\ (\Psi \circ \Phi)(f) =& \Psi \circ ((f \circ \pi_1) \times (f \circ \pi_2))\\ =&(f\circ \pi_1) \oplus (f \circ \pi_2)\\ =&f \end{split}$$$$

Is this correct? I believe I should be showing the inverses by looking at the images of specific elements in the modules themselves, to do this. What is a better/correct way to solve this problem?

The idea is correct, but there are some small issues. For example, how do you define $$f_1 \oplus f_2$$? Usually when you have $$f_1 : M_1 \to N_1$$, $$f_2 : M_2 \to N_2$$, the map $$f_1 \oplus f_2 : M_1 \oplus M_2 \to N_1 \oplus N_2$$ is defined by applying $$f_1$$, $$f_2$$ componentwise. Furthermore the compositions $$f \circ \pi_1, f \circ \pi_2$$ do not make sense, you want $$f \circ j_1, f \circ j_2$$, with $$j_1, j_2$$ the inclusions $$M \to M \oplus P, P \to M \oplus P$$.
Define $$\Phi: \mathrm{Hom}_R(M \oplus P, L) \to \mathrm{Hom}_R(M, L) \oplus \mathrm{Hom}_R(P, L)$$ by sending $$f \in \mathrm{Hom}_R(M \oplus P, L)$$ to the pair $$(f \circ j_1, f\circ j_2)$$.
Conversely, given $$f_1 \in \mathrm{Hom}_R(M, L)$$, $$f_2 \in \mathrm{Hom}_R(P, L)$$, define $$f : M \oplus P \to L$$ by $$f(m, p) = f_1(m) + f_2(p)$$, giving a homomorphism $$\Psi: \mathrm{Hom}_R(M, L) \oplus \mathrm{Hom}_R(P, L) \to \mathrm{Hom}_R(M \oplus P, L).$$