Unity Divides All Elements
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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 1
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.
$\blacksquare$
Proof 2
This is a special case of Unit of Integral Domain divides all Elements, as Unity is Unit.
Furthermore, from Unity and Negative form Subgroup of Units we also have that $-1_D$ is a unit of $D$.
Hence the result.
$\blacksquare$