Equality of Division Products

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


$\dfrac a c = \dfrac b d \iff a \circ d = b \circ c$

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


\(\ds \frac a c\) \(=\) \(\ds \frac b d\)
\(\ds \leadstoandfrom \ \ \) \(\ds a \circ c^{-1}\) \(=\) \(\ds b \circ d^{-1}\) Definition of Division Product
\(\ds \leadstoandfrom \ \ \) \(\ds a \circ c^{-1} \circ c \circ d\) \(=\) \(\ds b \circ d^{-1} \circ c \circ d\) Definition of Cancellable Element of $U_R$
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {a \circ d} \circ \paren {c^{-1} \circ c}\) \(=\) \(\ds \paren {b \circ c} \circ \paren {d^{-1} \circ d}\) Definition of Commutative Operation and Definition of Associative Operation
\(\ds \leadstoandfrom \ \ \) \(\ds a \circ d\) \(=\) \(\ds b \circ c\) Definition of Identity Element and Definition of Inverse Element