Odd Integers do not form Integral Domain

Theorem

Let $S$ denote the set of odd integers.

Then $\struct {S, +, \times}$ is not an integral domain.

Proof

Consider the odd integers $1$ and $3$.

We have that $1 + 3 = 4$.

But $4$ is not odd.

Thus addition on $\struct {S, +, \times}$ is not closed.

Hence $\struct {S, +, \times}$ is not even a ring, let alone an integral domain.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Exercises: Chapter $1$: Exercise $1 \ \text{(ii)}$