Element of Integral Domain Divides Zero

Theorem

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

Then every element of $D$ divides $0_D$:

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

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

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62.1$ Factorization in an integral domain: $\text{(i)}$