4. The first isomorphism theorem

For groups, recall the ``first isomorphism theorem'': If $\varphi: G \rightarrow H$, then im$\;\varphi \cong G/$ker$\varphi$. Or equivalently, if $\varphi: G \rightarrow H$ is a surjection with kernel $K$, then $H \cong G/K$.

This theorem is useful in examples. It also shows that the homomorphic images of a group $G$ are determined up to isomorphism by information internal to $G$. In particular, if $G$ is finite then up to isomorphism $G$ has only finitely many homomorphic images.

For algebras in general, the situation is the same:

4.1 Theorem (first isomorphism theorem). Let $\varphi: {\cal A} \rightarrow {\cal B}$ be a homomorphism. Then im$\; \varphi \cong A/$ker$\; \varphi$. Equivalently, if $\varphi: {\cal A} \rightarrow {\cal B}$ is a surjective homomorphism with kernel $\theta$, then ${\cal B} \cong {\cal A}/\theta$.

4.2 Corollary. The possible homomorphic images of ${\cal A}$ are determined up to isomorphism by the internal structure of ${\cal A}$.