Quotient Ring of Ring with Unity is Ring with Unity

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

If $$\left({R, +, \circ}\right)$$ is a ring with unity, then so is $$\left({R / J, +, \circ}\right)$$, 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, $$1_R \in J \Longrightarrow J = R$$. 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.