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

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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

From Left Cosets are Equal iff Product with Inverse in Subgroup, we have:

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

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

$\blacksquare$


Also see


Sources