Homomorphism theorem

Remarks

Proof idea. Compatibility of $\ker \alpha$ follows from $\alpha(f(a_1, \dots, a_n)) = f(\alpha(a_1), \dots, \alpha(a_n))$. Put $\beta(a/\theta) = \alpha(a)$; this is well defined because $\theta \subseteq \ker \alpha$, and it is a homomorphism because $\alpha$ and $\nu$ are. Uniqueness holds since $\nu$ is surjective, and $\beta(a/\theta) = \beta(b/\theta)$ means $(a, b) \in \ker \alpha$.

Example: the ring homomorphism $\Z \to \Z/n\Z$ has kernel congruence modulo $n$, so the quotient of $\Z$ by this congruence is isomorphic to $\Z/n\Z$. For groups, part 4 is the first isomorphism theorem.

Going deeper

For a student

Part 2 is the universal property of the quotient: maps out of $\mathbf{A}/\theta$ are the maps out of $\mathbf{A}$ that do not separate $\theta$-related elements. Part 1, together with $\ker \nu = \theta$, shows that congruences are exactly the kernels of homomorphisms. With the correspondence theorem it yields the other isomorphism theorems. It is basic in the proof of Birkhoff's HSP theorem, since it identifies the class operator $H$ with taking quotients.