Ideal is Additive Normal Subgroup
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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.
$\blacksquare$