Quotient Group of Ideal is Coset Space

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring.

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 $$x \in R$$.

The rule of addition of these cosets is: $$\left({x + J}\right) + \left({y + J}\right) = \left({x + y}\right) + J$$.

The identity of $$\left({R / J, +}\right)$$ is $$J$$ and for each $$x \in R$$, the inverse of $$x + J$$ is $$\left({-x}\right) + J$$.

Proof
From the definition of a ring, the additive group $$\left({R, +}\right)$$ is abelian and therefore normal.

Therefore the quotient group $$\left({R / J, +}\right)$$ is defined.

The rest follows directly from the definition of quotient group.