direct sum – Extending linear maps on modules.

Let $${M_i}_{iin mathbb{Z}}$$ be an index family of $$R$$-modules. Then we know that the direct sum $$bigoplus_{iin mathbb{Z}}M_i$$ is a submodule of a direct product $$prod_{iin mathbb{Z}}M_i$$. Let $$N$$ be an $$R$$-module and if $$f:bigoplus_{iin mathbb{Z}}M_ito N$$ is any linear map. Can we always extend $$f$$ to a linear map $$tilde{f}:prod_{iin mathbb{Z}}M_ito N .$$ That is, $$tilde{f}$$ is a linear map such that $$tilde{f}|_{bigoplus_{iin mathbb{Z}}M_i} = f.$$ I know in vector spaces we can always construct $$tilde{f}$$ with the help of a basis, but in general modules how do we proceed?

Secondly, if some $$f$$ has an extention to $$tilde{f}$$, is $$tilde{f}$$ unique?