Definition:Division Product

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

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers