Characteristic of Ring of Integers Modulo Prime

Theorem

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

The characteristic of $\left({\Z_p, +, \times}\right)$ is $p$.

Proof

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

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

The result follows.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.17$: Example $23$