Subring of Integers is Ideal

Theorem
Let $\struct {\Z, +}$ be the additive group of integers.

Every subring of $\struct {\Z, +, \times}$ is an ideal of the ring $\struct {\Z, +, \times}$.

Proof
Follows directly from:
 * Subrings of Integers are Sets of Integer Multiples

and:
 * Subgroup of Integers is Ideal.