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 $1_R$.

Thus:
 * $\paren {-z}^{-1} = -\paren {z^{-1} }$