Even Integers do not form Integral Domain
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Theorem
Let $2 \Z$ be the set of even integers.
Then $\struct {2 \Z, +, \times}$ is not an integral domain.
Proof
From Integer Multiples form Commutative Ring, $\struct {2 \Z, +, \times}$ is a commutative ring.
As $2 \ne 1$, we also have from Integer Multiples form Commutative Ring that $\struct {2 \Z, +, \times}$ has no unity.
Hence by definition it is not an integral domain.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Exercises: Chapter $1$: Exercise $1 \ \text{(i)}$