Quotient Ring is Ring

Theorem
Let $\struct {R, +, \circ}$ be a ring.

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

Let $\struct {R / J, +, \circ}$ 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 $\circ$
Now to prove that $\struct {R / J, +, \circ}$ is a ring, proceed by verifying the ring axioms in turn:

From:
 * Ideal is Additive Normal Subgroup
 * The definition of a quotient group
 * Quotient Group is Group

it follows that $\struct {R / J, +}$ is a group.

By definition of $\circ$ in $R / J$, it follows that $\struct {R / J, \circ}$ is closed.

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

Distributivity can be deduced from the fact that $\circ$ is distributive on $R$:

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