Quotient Ring is Ring

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

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

Let $\left({R / J, +, \circ}\right)$ be the quotient ring of $R$ by $J$.

Then $R / J$ is also a ring.

Proof
First, it is to be shown that $+$ and $\circ$ are in fact well-defined operations on $R / J$.

Well-definition of $+$
For $+$, it follows from Ideal is Additive Normal Subgroup that $J$ is a normal subgroup of $R$ under $+$.

Thus, the quotient group $\left({R / J, +}\right)$ is defined, and as a Quotient Group is a Group, $+$ is well-defined.

Well-definition of $\circ$
Now to prove that $\left({R / J, +, \circ}\right)$ is a ring, proceed by verifying the ring axioms in turn:

A: Addition forms a Group
As mentioned before, this follows from combining Ideal is Additive Normal Subgroup and Quotient Group is a Group.

M0: Closure of Ring Product
By definition of $\circ$ in $R / J$, it follows that $\left({R / J, \circ}\right)$ is closed.

M1: Associativity of Ring Product
Associativity can be deduced from the fact that $\circ$ is associative on $R$:

D: Distributivity of Ring Product over Addition
Distributivity can be deduced from the fact that $\circ$ is distributive on $R$:

Having verified all of the ring axioms, it follows that $\left({R / J, +, \circ}\right)$ is a ring.