Definition:Division Product

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Definition

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

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


Then we define the following notation:

$\forall x \in U_R, y \in R$, we have:
$\dfrac y x := y \circ \paren {x^{-1} } = \paren {x^{-1} } \circ y$


$\dfrac y x$ is a division product, and $\dfrac y x$ is voiced $y$ divided by $x$.


We also write (out of space considerations) $y / x$ for $\dfrac y x$.


This notation is usually used when $\struct {R, +, \circ}$ is a field.


Caution

We do not usually use this notation for a ring (with unity) which is not commutative, as it would not be straightforward to determine whether $\dfrac y x$ means $y \circ \paren {x^{-1} }$ or $\paren {x^{-1} } \circ y$.


Sources