Characteristic of Integral Domain is Zero or Prime

Theorem

Let $\struct {D, +, \circ}$ be an integral domain.

Let $\Char D$ be the characteristic of $D$.

Then $\Char D$ is either $0$ or a prime number.

Proof

By definition, an integral domain has no proper zero divisors.

If $\struct {D, +, \circ}$ is finite, then from Characteristic of Finite Ring with No Zero Divisors, $\Char D$ is prime.

On the other hand, suppose $\struct {D, +, \circ}$ is not finite.

Then there are no $x, y \in D, x \ne 0 \ne y$ such that $x + y = 0$.

Thus it follows that $\Char D$ is $0$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): characteristic: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): characteristic: 2.