How to show that $\mathbb{Z}_5[i] = \{a + bi:a,b\in\mathbb{Z}_5 \}$ is a ring.

Question

A set with modular arithmetic is given: $$(\mathbb{Z}_5[i], +_5, \cdot_5)$$ $$\mathbb{Z}_5[i] = \{a + bi: a,b \in \mathbb{Z}_5\}$$ I am to show that this set is a ring. Of course I would like to show that is it a subring but I don't know which one could be the "bigger" ring.

Answer 1

$\mathbb Z_5[i]\cong \mathbb Z_5[x]/(x^2+1)\cong \mathbb Z_5[x]/(x-2)(x+2) \cong \mathbb Z_5 \times \mathbb Z_5 $, so $\mathbb Z_5[i]$ is a ring.

Answer 2

There are two natural interpretations for $\mathbb{Z}_5[i]$:

• $\mathbb{Z}[i]/(5)$

• $\mathbb{Z}_5[x]/(x^2+1)$

They are actually the same, because these two rings are isomorphic to $$ \mathbb{Z}[x]/(5,x^2+1) $$ In all cases, the ring operations are the natural ones induced by the quotient map.