Ring of Integers Modulo Prime is Integral Domain/Proof 1

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Corollary to Ring of Integers Modulo Prime is Field

Let $m \in \Z: m \ge 2$.

Let $\struct {\Z_m, +, \times}$‎ be the ring of integers modulo $m$.


Then:

$m$ is prime

if and only if:

$\struct {\Z_m, +, \times}$ is an integral domain.


Proof

We have that a Field is Integral Domain.

We also have that a Finite Integral Domain is Galois Field.

The result follows from Ring of Integers Modulo Prime is Field.

$\blacksquare$