Unity Divides All Elements

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain whose unity is $1_D$.

Then unity is a divisor of every element of $D$:
 * $\forall x \in D: 1_D \mathop \backslash x$

Also:


 * $\forall x \in D: -1_D \mathop \backslash x$

Proof 1
The element $1_D$ is the unity of $\left({D, +, \circ}\right)$, and so:
 * $1_D \in D: x = 1_D \circ x$

Similarly, from Product of Ring Negatives:


 * $-1_D \in D: x = \left({-1_D}\right) \circ \left({-x}\right)$

The result follows from the definition of divisor.

Proof 2
This is a special case of Every Unit Divides Every Element, as Unity is a Unit.

Furthermore, from Unity and Negative Subgroup of Units we also have that $-1_D$ is a unit of $D$.

Hence the result.