Unity Divides All Elements/Proof 1

Theorem

Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$.

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

$\forall x \in D: 1_D \divides x$

Also:

$\forall x \in D: -1_D \divides x$

Proof

The element $1_D$ is the unity of $\struct {D, +, \circ}$, and so:

$1_D \in D: x = 1_D \circ x$

Similarly, from Product of Ring Negatives:

$-1_D \in D: x = \paren {-1_D} \circ \paren {-x}$

The result follows from the definition of divisor.

$\blacksquare$