Confusion with integral domains

Question

As I understand it, an integral domain is a commutative ring with no zero divisors. I have seen examples of finite integral domains, which I will call $D$ (i.e. $J_p$), but how can this be? A simple corollary to Lagrange's Theorem is that for any $d\in D$, $d*\mid D\mid=0$. What am I missing?

Answer 1

Your confusion is that $|D|$ doesn’t necessarily belong to $D$. For $D$ to be an integral domain any product of two non-zero elements in $D$ must be non-zero. For example $\mathbb Z_5$, which is an integral domain whose order is $5$, but $5$ is not contained in $\mathbb Z_5$.