# 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 $J \subsetneq R$.

$1_R \in J \implies J = R$

So $1_R \notin J$.

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

Now let $x \in R$.

 $\ds \paren {1_R + J} \circ \paren {x + J}$ $=$ $\ds 1_R \circ x + J$ Definition of $\circ$ in $R / J$ $\ds$ $=$ $\ds x + J$ Definition of Unity of Ring $\ds$ $=$ $\ds x \circ 1_R + J$ Definition of Unity of Ring $\ds$ $=$ $\ds \paren {x + J} \circ \paren {1_R + J}$ Definition of $\circ$ in $R / J$

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.

$\blacksquare$