Subgroup of Integers is Ideal

Theorem

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

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

Corollary

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

Proof

Let $H$ be a subgroup of $\struct {\Z, +}$.

Let $n \in \Z, h \in H$.

Then from the definition of cyclic group and Negative Index Law for Monoids:

$n h = n \cdot h \in \gen h \subseteq H$

The result follows.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.2$
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Example $34$