Quotient Theorem for Epimorphisms

Theorem
Let $\struct {S, \oplus}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \oplus} \to \struct {T, *}$ be an epimorphism.

Let $\RR_\phi$ be the equivalence induced by $\phi$.

Let $S / \RR_\phi$ be the quotient of $S$ by $\RR_\phi$.

Let $q_{\RR_\phi}: S \to S / \RR_\phi$ be the quotient mapping induced by $\RR_\phi$.

Let $\struct {S / \RR_\phi}, {\oplus_{\RR_\phi} }$ be the quotient structure defined by $\RR_\phi$.

Then:


 * The induced equivalence $\RR_\phi$ is a congruence relation for $\oplus$
 * There exists a unique isomorphism $\psi: \struct {S / \RR_\phi}, {\oplus_{\RR_\phi} } \to \struct {T, *}$ which satisfies $\psi \circ q_{\RR_\phi} = \phi$

where $\circ$ denotes composition of mappings.

Also known as
Some authors call this the Factor Theorem for Epimorphisms.