Definition:Quotient Ring

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

Let $$\equiv$$ be an equivalence relation on $$R$$ compatible with both $$\circ$$ and $$+$$.

Then $$\left({R / \equiv, +_\equiv, \circ_\equiv}\right)$$ is a ring, where $$R / \equiv$$is the quotient set of $R$ by $\equiv$.

The ring $$\left({R / \equiv, +_\equiv, \circ_\equiv}\right)$$ is the quotient ring of $$R$$ and $$\equiv$$.

Proof
This follows from the fact that, for a congruence $$\equiv$$, the quotient mapping from $$R$$ to $$R / \equiv$$ is an epimorphism.

As $$\equiv$$ is an equivalence relation on $$R$$ compatible with both $$\circ$$ and $$+$$, it is therefore a congruence on $$R$$ for both operations.