Congruence Relation and Ideal are Equivalent

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

Let $\EE$ be an equivalence relation on $R$ compatible with both $\circ$ and $+$, that is, a congruence relation on $R$.

Let $J = \eqclass {0_R} \EE$ be the equivalence class of $0_R$ under $\EE$.

Then:
 * $(1a): \quad J = \eqclass {0_R} \EE$ is an ideal of $R$
 * $(2a): \quad$ The equivalence defined by the quotient ring $R / J$ is $\EE$ itself.

Similarly, let $J$ be an ideal of $R$.

Then:
 * $(1b): \quad J$ induces a congruence relation $\EE_J$ on $R$
 * $(2b): \quad$ The ideal of $R$ defined by $\EE_J$ is $J$ itself.

Part $(1a)$
This is shown on Congruence Relation on Ring induces Ideal.

Part $(2a)$
This is shown on Ideal induced by Congruence Relation defines that Congruence.

Part $(1b)$
This is shown on Ideal induces Congruence Relation on Ring.