Let $R$ be a ring with identity, let $M$ be a free left $R$-module with basis $(e_i)_{i \in I}$, and let $N$ be a left $R$-module. For every family $(n_i)_{i \in I}$ of elements of $N$, there exists a unique left $R$-[module homomorphism](/page/Module%20Homomorphism) $f: M \to N$ such that $f(e_i) = n_i$ for every $i \in I$.