# Ideal induced by Congruence Relation defines that Congruence

## Theorem

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

Let $\EE$ be a congruence relation on $R$.

Let $J = \eqclass {0_R} \EE$ be the ideal induced by $\EE$.

Then the equivalence defined by the coset space $\struct {R, +} / \struct {J, +}$ is $\EE$ itself.

## Proof

Let $J = \eqclass {0_R} \EE$.

From Congruence Relation on Ring induces Ideal, we have that $J$ is an ideal of $R$.

From Ideal is Additive Normal Subgroup, we have that $\struct {J, +}$ is a normal subgroup of $\struct {R, +}$.

From Normal Subgroup induced by Congruence Relation defines that Congruence, the equivalence defined by the partition $\struct {R, +} / \struct {J, +}$ is $\EE$.

As $\EE$ was the congruence relation on $R$ that was originally posited, we already know that it is compatible with $\circ$.

Thus the equivalence defined by $J$ is the same congruence relation on $R$ that gave rise to $J$ to start with.

Hence the result.

$\blacksquare$

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorems $22.2$