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 defined by $$J$$.

We define the quotient ring product on the set of elements of $$\left({R / J, +, \circ}\right)$$ as:

$$\left({x + J}\right) \circ \left({y + J}\right) = x \circ y + J$$

The quotient ring product is well-defined:

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

Proof
From Sundry Coset Results: Theorem 2, we have:


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

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: