Definition:Quotient Ring

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

Let $$J$$ be an ideal of $$R$$.

From Quotient Group of Ideal is Coset, each element of $$\left({R / J, +}\right)$$ is of the form $$x + J = \left\{{x + j: j \in J}\right\}$$ for some $$x \in R$$.

Let addition be defined on $$\left({R / J, +, \circ}\right)$$ as here:
 * $$\left({x + J}\right) + \left({y + J}\right) = \left({x + y}\right) + J$$

and ring product be as defined here:
 * $$\left({x + J}\right) \circ \left({y + J}\right) = x \circ y + J$$

Let $$\mathcal E_J$$ be the congruence relation induced by $$J$$.

Then $$\left({R / J, +, \circ}\right)$$ is a ring called the quotient ring of $$R$$ and $$\mathcal E_J$$.

Also see
In Quotient Ring is an Ideal it is proved not only that $$\left({R / J, +, \circ}\right)$$ is a ring, but also that it is an ideal of $$R$$.