Product of Division Products

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Theorem

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

Let $\struct {U_R, \circ}$ be the group of units of $\struct {R, +, \circ}$.


Let $a, b \in R, c, d \in U_R$.

Then:

$\dfrac a c \circ \dfrac b d = \dfrac {a \circ b} {c \circ d}$

where $\dfrac x z$ is defined as $x \circ \paren {z^{-1} }$, that is, $x$ divided by $z$.


Proof

\(\displaystyle \frac a c \circ \frac b d\) \(=\) \(\displaystyle a \circ c^{-1} \circ b \circ d^{-1}\) Definition of Division Product
\(\displaystyle \) \(=\) \(\displaystyle \paren {a \circ b} \circ \paren {d^{-1} \circ c^{-1} }\) Definition of Commutative Operation
\(\displaystyle \) \(=\) \(\displaystyle \paren {a \circ b} \circ \paren {c \circ d}^{-1}\) Inverse of Product
\(\displaystyle \) \(=\) \(\displaystyle \frac {a \circ b} {c \circ d}\) Definition of Division Product

$\blacksquare$


Sources