Unity Divides All Elements/Proof 2

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

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.

$\blacksquare$


Sources

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