The Gaussian integers $\mathbb{Z}[i]$

Ring. Closed under addition/subtraction and multiplication: $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$ with integer coefficients. Commutative with unit $1$.

Integral domain. $\mathbb{Z}[i]\subseteq\mathbb{C}$, which is a field, so no zero divisors.

Norm. $N(z)=z\bar z=a^2+b^2\in\mathbb{Z}_{\ge0}$, multiplicative: $N(zw)=N(z)N(w)$.

Units. $u$ is a unit iff $N(u)=1$ (since $u^{-1}$ would have norm $1/N(u)$, and this must be an integer). Solutions of $a^2+b^2=1$: $(\pm1,0),(0,\pm1)$. So units are $\pm1,\pm i$.

Conclusion. $\mathbb{Z}[i]$ is an integral domain with units $\{\pm1,\pm i\}$.