Quotient Ring is Ring/Quotient Ring Addition 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 $+$ is well-defined on $R / J$, that is:


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

Proof
From Ideal is Additive Normal Subgroup that $J$ is a normal subgroup of $R$ under $+$.

Thus, the quotient group $\left({R / J, +}\right)$ is defined, and as a Quotient Group is a Group, $+$ is well-defined.