# Quotient Ring is Ring/Quotient Ring Product is Well-Defined

## 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 of $R$ by $J$.

Then $\circ$ is well-defined on $R / J$, that is:

$x_1 + J = x_2 + J, y_1 + J = y_2 + J \implies x_1 \circ y_1 + J = x_2 \circ y_2 + J$

## Proof

 $\displaystyle x_1 + J$ $=$ $\displaystyle x_2 + J$ $\displaystyle \leadsto \ \$ $\displaystyle x_1 + \paren {-x_2}$ $\in$ $\displaystyle J$

and:

 $\displaystyle y_1 + J$ $=$ $\displaystyle y_2 + J$ $\displaystyle \leadsto \ \$ $\displaystyle y_1 + \paren {-y_2}$ $\in$ $\displaystyle J$

Hence from the definition of ideal:

 $\displaystyle \paren {x_1 + \paren {-x_2} } \circ y_1$ $\in$ $\displaystyle J$ $\displaystyle x_2 \circ \paren {y_1 + \paren {-y_2} }$ $\in$ $\displaystyle J$

Thus:

 $\displaystyle \paren {x_1 + \paren {-x_2} } \circ y_1 + x_2 \circ \paren {y_1 + \paren {-y_2} }$ $\in$ $\displaystyle J$ as $\struct {J, +}$ is a group $\displaystyle \leadsto \ \$ $\displaystyle x_1 \circ y_1 + \paren {-\paren {x_2 \circ y_2} }$ $\in$ $\displaystyle J$ Various ring properties $\displaystyle \leadsto \ \$ $\displaystyle x_1 \circ y_1 + J$ $=$ $\displaystyle x_2 \circ y_2 + J$ Left Cosets are Equal iff Product with Inverse in Subgroup

$\blacksquare$