Ideal induces Congruence Relation on Ring

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

Let $J$ be an ideal of $R$

Then $J$ induces a congruence relation $\mathcal E_J$ on $R$ such that $\left({R / J, +, \circ}\right)$ is a quotient ring.

Proof
Let $x + \left({-x'}\right), y + \left({-y'}\right) \in J$. Then:


 * $x \circ y + \left({- x' \circ y'}\right) = \left({x + \left({-x'}\right)}\right) \circ y + x' \circ \left({y + \left({-y'}\right)}\right) \in J$

Hence the result.