Ideal induced by Congruence Relation defines that Congruence

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\mathcal E$ be a congruence relation on $R$.

Let $J = \left[\!\left[{0_R}\right]\!\right]_\mathcal E$ be the ideal induced by $\mathcal E$.

Then the equivalence defined by the coset space $\left({R, +}\right) / \left({J, +}\right)$ is $\mathcal E$ itself.

Proof
Let $J = \left[\!\left[{0_R}\right]\!\right]_\mathcal E$.

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 $\left({J, +}\right)$ is a normal subgroup of $\left({R, +}\right)$.

From Congruence Relation arises from Normal Subgroup, the equivalence defined by the partition $\left({R, +}\right) / \left({J, +}\right)$ is $\mathcal E$.

Also see

 * Quotient Ring is a Ring