# Congruence Relation on Ring induces Ring

## Theorem

Let $\struct {R, +, \circ}$ be a ring.

Let $\EE$ be a congruence relation on $R$ for both $+$ and $\circ$.

Let $R / \EE$ be the quotient set of $R$ by $\EE$.

Let $+_\EE$ and $\circ_\EE$ be the operations induced on $R / \EE$ by $+$ and $\circ$ respectively.

Then $\struct {R / \EE, +_\EE, \circ_\EE}$ is a ring.

## Proof

Let $q_\EE$ be the quotient mapping from $\struct {R, +, \circ}$ to $\struct {R / \EE, +_\EE, \circ_\EE}$.

From Quotient Mapping on Structure is Epimorphism:

- $q_\EE: \struct {R, +} \to \struct {R / \EE, +_\EE}$ is an epimorphism

- $q_\EE: \struct {R, \circ} \to \struct {R / \EE, \circ _\EE}$ is an epimorphism.

As the morphism property holds for both $+$ and $\circ$, it follows that $q_\EE: \struct {R, +, \circ} \to \struct {R / \EE, +_\EE, \circ_\EE}$ is also an epimorphism.

From Epimorphism Preserves Rings, it follows that $\struct {R / \EE, +_\EE, \circ_\EE}$ is a ring.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.1$