Quotient Group of Ideal is Coset Space

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 $$\left({R / J, +}\right)$$ be the Quotient Group of $$\left({R, +}\right)$$ by $$\left({J, +}\right)$$.

Then each element of $$\left({R / J, +}\right)$$ is a coset of $$J$$ in $$R$$, that is, is of the form $$x + J = \left\{{x + j: j \in J}\right\}$$ for some $$x \in R$$.

Proof
Follows directly from Quotient Group.