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.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $23$