Unity and Negative form Subgroup of Units

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Theorem

Let $\struct {R, +, \circ}$ be a ring with unity.

Then:

$\struct {\set {1_R, -1_R}, \circ} \le U_R$

That is, the set consisting of the unity and its negative forms a subgroup of the group of units.


Proof

From Unity is Unit:

$1_R \in U_R$

It remains to be shown that $-1_R \in U_R$.

From Ring Axiom $\text A$: Addition forms an Abelian Group, $\struct {R, +}$ is an abelian group.

Therefore:

$1_R \in R \implies -1_R \in R$.

From Product of Ring Negatives:

$-1_R \circ -1_R = 1_R \circ 1_R = 1_R$

Thus $-1_R$ has a ring product inverse (itself) and therefore $-1_R \in U_R$.


We have:

\(\ds 1_R \circ 1_R\) \(=\) \(\ds 1_R\)
\(\ds 1_R \circ -1_R\) \(=\) \(\ds -1_R\)
\(\ds -1_R \circ 1_R\) \(=\) \(\ds -1_R\)
\(\ds -1_R \circ -1_R\) \(=\) \(\ds 1_R\)

This exhausts all the ways we can form a ring product between $1_R$ and $-1_R$.


That is:

$\forall x, y \in \set {1_R, -1_R}: x \circ y \in \set {1_R, -1_R}$

Hence the result from the Two-Step Subgroup Test.

$\blacksquare$