Quotient Ring of Ring with Unity is Ring with Unity

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

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

Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$.

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

Proof
Let $\struct {R, +, \circ}$ be a ring with unity.

First, let $J \subsetneq R$.

By Ideal of Unit is Whole Ring: Corollary:
 * $1_R \in J \implies J = R$

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:
 * $R / J = R / R = \set {0_{R / R} }$

which is the null ring.

Also see

 * Ring Without Unity may have Quotient Ring with Unity