# Universal Property of Free Modules

Jump to navigation
Jump to search

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

This article needs to be tidied.Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Tidy}}` from the code. |

## Theorem

Let $R$ be a ring.

Let $M$ be a free $R$-module with basis $\{e_i\mid i\in I\}$.

Let $N$ be an $R$-module.

Let $\{n_i\mid i\in I\}$ be a family of elements of $N$.

Then there exists a unique $R$-module homomorphism that maps $e_i$ to $n_i$ for all $i\in I$.

## Proof

Combine Free Module is Isomorphic to Free Module Indexed by Set and Universal Property of Free Module on Set.

This article contains statements that are justified by handwavery.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Handwaving}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |