Let $R$ be a unital ring with additive identity $0_R$ and multiplicative identity $1_R$, let $N$ be a unital left $R$-module with additive identity $0_N$, and let $R^n$ be the free left $R$-module with standard basis $e_1,\ldots,e_n$. For every choice of elements $y_1,\ldots,y_n\in N$, there exists a unique $R$-[module homomorphism](/page/Module%20Homomorphism) $f:R^n\to N$ such that, for every $i\in\{1,\ldots,n\}$,

\begin{align*} f(e_i)=y_i. \end{align*}