Ideal is Additive Normal Subgroup

Theorem

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

Let $J$ be an ideal of $R$.

Then $\left({J, +}\right)$ is a normal subgroup of $\left({R, +}\right)$.

Proof

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

By definition of a ring, $\left({R, +}\right)$ is abelian.

The result follows from Subgroup of Abelian Group is Normal.

$\blacksquare$