# Definition:Ring Extension

*This page is about ring extensions. For other uses, see Extension.*

This page has been identified as a candidate for refactoring of medium complexity.In particular: There are statements here which are not part of the actual definition which need to be extracted into theorem pages. Additionally, two definitionsUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.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 `{{Refactor}}` from the code. |

## Definition

Let $R$ and $S$ be commutative rings with unity.

Let $\phi : R \to S$ be a ring monomorphism.

Then $\phi : R \to S$ is a **ring extension** of $R$.

Alternatively, we can define $S$ to be a ring extension of $R$ if $R$ is a subring of $S$ (provided we insist that a subring inherits the multiplicative identity from its parent ring).

These definitions are equivalent up to isomorphism, for if $R \subseteq S$ is a subring, then the identity $\operatorname{id} : R \to S$ is a monomorphism.

Conversely if $\phi : R \to S$ is a monomorphism, then $\operatorname{im}\phi \subseteq S$ is a subring of $S$.

Moreover by Surgery for Rings, we can find a ring $T$, isomorphic to $S$, that contains $R$ as a subring.

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: The link between these two definitions needs to be made clearer. Believe that Ring Homomorphism Preserves Subrings could be used as the basis for this. Needs more thought.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |