Quotient Ring of Ring with Unity is Ring with Unity

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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 $x \in R$.

\(\displaystyle \paren {1_R + J} \circ \paren {x + J}\) \(=\) \(\displaystyle 1_R \circ x + J\) Definition of $\circ$ in $R / J$
\(\displaystyle \) \(=\) \(\displaystyle x + J\) Definition of Unity of Ring
\(\displaystyle \) \(=\) \(\displaystyle x \circ 1_R + J\) Definition of Unity of Ring
\(\displaystyle \) \(=\) \(\displaystyle \paren {x + J} \circ \paren {1_R + J}\) 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.

$\blacksquare$


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