Definition:Ring Extension

Definition
Let $R$ and $S$ be rings.

Let $\phi : R \to S$ be an injective homomorphism of rings.

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 $S$.

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

Conversely if $\phi : R \to S$ is an injective homomorphism, 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.

Also See

 * Field extension