# Quotient Ring of Commutative Ring is Commutative

## Theorem

Let $\struct {R, +, \circ}$ be a ring 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 $J$.

If $\struct {R, +, \circ}$ is a commutative ring, then so is $\struct {R / J, +, \circ}$.

## Proof

Let $\struct {R, +, \circ}$ be a commutative ring

That means $\circ$ is commutative on $R$.

Thus:

 $\displaystyle \forall x, y \in R: \ \$ $\displaystyle \paren {x + J} \circ \paren {y + J}$ $=$ $\displaystyle x \circ y + J$ Definition of $\circ$ in $R / J$ $\displaystyle$ $=$ $\displaystyle y \circ x + J$ Commutativity of $\circ$ $\displaystyle$ $=$ $\displaystyle \paren {y + J} \circ \paren {x + J}$ Definition of $\circ$ in $R / J$

$\blacksquare$