Unity Divides All Elements

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

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

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


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

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.


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.