Product of Negative with 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:

$(1): \quad \forall x \in R: -\paren {x \circ z^{-1} } = \paren {-x} \circ z^{-1} = x \circ \paren {\paren {-z}^{-1} }$

$(2): \quad \forall x \in R: -\paren {z^{-1} \circ x} = z^{-1} \circ \paren {-x} = \paren {\paren {-z}^{-1} } \circ x$

$\text {(1)}: \quad$

$\ds -\paren {x \circ z^{-1} }$

$=$

$\ds \paren {-x} \circ z^{-1}$

$\ds $

$=$

$\ds x \circ \paren {-\paren {z^{-1} } }$

$\ds $

$=$

$\ds x \circ \paren {\paren {-z}^{-1} }$

$\Box$

$\text {(2)}: \quad$

$\ds -\paren {z^{-1} \circ x}$

$=$

$\ds z^{-1} \circ \paren {-x}$

$\ds $

$=$

$\ds \paren {-\paren {z^{-1} } } \circ x$

$\ds $

$=$

$\ds \paren {\paren {-z}^{-1} } \circ x$

$\blacksquare$

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