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.

$\blacksquare$

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.3$: Some properties of subrings and ideals