Definition:Quotient Ring

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

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

Let addition be defined on $$\left({R / J, +, \circ}\right)$$ as here, and ring product be as defined here.

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 $$R$$.