# Element of Integral Domain Divides Zero

## Theorem

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$.

Then every element of $D$ is a divisor of zero:

$\forall x \in D: x \divides 0_D$

## Proof

By definition, an integral domain is a ring.

So, from Ring Product with Zero:

$\forall x \in D: 0_D = x \circ 0_D$.

The result follows from the definition of divisor.

$\blacksquare$