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

Theorem
Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$ and whose unity is $1_R$.

Let $J$ be an ideal of $R$.

Let $\left({R / J, +, \circ}\right)$ 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:


 * $x_1 + J = x_2 + J \implies x_1 + \left({-x_2}\right) \in J$
 * $y_1 + J = y_2 + J \implies y_1 + \left({-y_2}\right) \in J$

hence from the definition of Ideal:


 * $\left({x_1 + \left({-x_2}\right)}\right) \circ y_1 \in J$
 * $x_2 \circ \left({y_1 + \left({-y_2}\right)}\right) \in J$

Thus:

Also see

 * Ideal induces Congruence Relation on Ring