Definition:Division Product

Definition

Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Let $\left({U_R, \circ}\right)$ be the group of units of $\left({R, +, \circ}\right)$.

Then we define the following notation:

$\forall x \in U_R, y \in R$, we have:

$\dfrac y x := y \circ \left({x^{-1}}\right) = \left({x^{-1}}\right) \circ y$

$\dfrac y x$ 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 $\left({R, +, \circ}\right)$ 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 $y \circ \left({x^{-1}}\right)$ or $\left({x^{-1}}\right) \circ y$.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.2$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 23$