Subring of Integers is Ideal

Theorem
Let $\left({\Z, +}\right)$ be the additive group of integers.

Every subring of $\left({\Z, +, \times}\right)$ is an ideal of the ring $\left({\Z, +, \times}\right)$.

Proof
Follows directly from Subrings of Integers and Subgroup of Integers is Ideal.