Can every map from a summand be uniquely determined by the canonical projection

Question

Let $A$ and $B$ be two $R$-modules, $C= A \oplus B$ and $f:M \longrightarrow A$ be an $R$-module homomorphism. For every $R$-homomorphism $f$ and every module $A$ can we guarantee that there is a unique map $g:M \longrightarrow C$ such that $f=\pi_A \circ g$ , where $\pi_A:C \longrightarrow A$ is the canonical projection?

Answer 1

No. Take $R=A=B=M=\mathbb{Z}$, $f=id$, and $g_1(x)=(x,0)$ the inclusion of $\mathbb{Z}$ into the first coordinate and $g_2(x)=(x,x)$ the diagonal inclusion. Then $\pi_1 g_i=f$ for both $i=1,2$.

Answer 2

Direct sums agree with direct product when there are only finitely many indices, and in this case, the product comes equipped with two projections. You need both of them to guarantee uniqueness.