Definition:Quotient Ring

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

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

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

Let $R / \mathcal E_J$ be the quotient structure defined by $\mathcal E_J$.

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:
 * $\left({x + J}\right) \circ \left({y + J}\right) = x \circ y + J$

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

Also known as
This is also sometimes called a factor ring.

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 $R$.