Element of Integral Domain Divides Zero

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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.

$\blacksquare$


Sources