Odd Integers do not form Integral Domain
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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 $1 \ \text{(ii)}$