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 an abelian group.

By the definition of an ideal, $\left({J, +}\right)$ is a subgroup of $\left({R, +}\right)$.

From Subgroup of Abelian Group is Normal it follows that $\left({J, +}\right)$ is a normal subgroup of $\left({R, +}\right)$.

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

The rest follows directly from the definition of quotient group.