Quotient Ring of Ring with Unity is Ring with Unity

Theorem
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

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

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

Then $\left({R / J, +, \circ}\right)$ is a ring with unity, and its unity is $1_R + J$.

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


 * First, let $J \subset R$, i.e. $J \ne R$.

By Ideal of Unit is Whole Ring, $1_R \in J \implies J = R$. So $1_R \notin J$.

Thus $1_R + J \ne J$, so $1_R + J \ne 0_{R/J}$.

Now let $x \in R$.

Thus $R / J$ has a unity, and that unity is $1_R + J$.


 * Now suppose $J = R$.

Then $1_R + J = J$ and therefore $1_R = 0_R$.

The only ring to have $1_R = 0_R$ is the null ring.

This is appropriate, because $R / J = R / R = \left\{{0_{R/R}}\right\}$ which is the null ring.