# Definition:Ring Extension

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

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## 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.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. |