Congruence Relation on Ring induces Ring

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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 Canonical 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