# Definition:Ring 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. A part of this page has to be extracted as a theorem: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.