\begin{definition}[Definition:Ring Extension]
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.
{{ExtractTheorem|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.}}
\end{definition}