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


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

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


First we demonstrate the operation has the specified property:

\(\ds \frac a b + \frac c d\) \(=\) \(\ds a \circ b^{-1} + c \circ d^{-1}\) Definition of Division Product
\(\ds \) \(=\) \(\ds a \circ b^{-1} \circ d \circ d^{-1} + c \circ d^{-1} \circ b \circ b^{-1}\) Definition of Inverse Element and Definition of Identity Element under $\circ$
\(\ds \) \(=\) \(\ds \paren {a \circ d} \circ \paren {d^{-1} \circ b^{-1} } + \paren {b \circ c} \circ \paren {d^{-1} \circ b^{-1} }\) Definition of Commutative Operation
\(\ds \) \(=\) \(\ds \paren {a \circ d + b \circ c} \circ \paren {b \circ d}^{-1}\) Definition of Distributive Operation $\circ$ over $+$
\(\ds \) \(=\) \(\ds \frac {a \circ d + b \circ c} {b \circ d}\) Definition of Division Product

Notice that this works only if $\struct {R, +, \circ}$ is commutative.