Finite ring

Remarks

For each integer $n \ge 1$, the ring $\Z/n\Z$ of integers modulo n is a finite ring of order $n$, since there are exactly $n$ residue classes. The ring $M_2(\F_2)$ of $2 \times 2$ matrices over the finite field $\F_2$ is a finite ring of order $2^4 = 16$, and it is not commutative. The ring $\Z$ of integers is not a finite ring, because it has infinitely many elements. Every finite integral domain is a field: multiplication by a nonzero element is injective, hence bijective on a finite set, so every nonzero element has an inverse.