Negative of Product Inverse

Theorem

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

Let $z \in U_R$, where $U_R$ is the set of units.

Then:

$\paren {-z}^{-1} = -\paren {z^{-1} }$

where $z^{-1}$ is the ring product inverse of $z$.

Proof

Let the unity of $\struct {R, +, \circ}$ be $1_R$.

\(\ds \paren {-\paren {z^{-1} } } \circ \paren {-z}\)

\(=\)

\(\ds z^{-1} \circ z\)

Product of Ring Negatives

\(\ds \)

\(=\)

\(\ds 1_R\)

Inverse under $\circ$

\(\ds \)

\(=\)

\(\ds z \circ z^{-1}\)

Inverse under $\circ$

\(\ds \)

\(=\)

\(\ds \paren {-z} \circ \paren {-\paren {z^{-1} } }\)

Product of Ring Negatives

Thus:

$\paren {-z}^{-1} = -\paren {z^{-1} }$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.7 \ (1)$