# Characteristic of Ring of Integers Modulo Prime

## Theorem

Let $\struct {\Z_p, +, \times}$ be the ring of integers modulo $p$, where $p$ is a prime number.

The characteristic of $\struct {\Z_p, +, \times}$ is $p$.

## Proof

From Ring of Integers Modulo Prime is Field we have that $\struct {\Z_p, +, \times}$ is a field.

So Characteristic of Finite Ring with No Zero Divisors applies, and so the characteristic of $\struct {\Z_p, +, \times}$ is prime.

The result follows.

$\blacksquare$