Definition:Integral Multiple

Definition

Rings and Fields

Let $\left({F, +, \times}\right)$ be a ring or a field.

Let $a \in F$.

Let $n \in \Z$ be an integer.

Then $n \cdot a$ is an integral multiple of $a$ where $n \cdot a$ is defined as in Powers of Ring Elements:

$n \cdot a = \begin{cases} 0_F & : n = 0 \\ a & : n = 1 \\ \left({\left({n - 1}\right) \cdot a}\right) + a & : n > 1 \\ \left|{n}\right| \cdot \left({-a}\right) & : n < 0 \\ \end{cases}$

where $\left|{n}\right|$ is the absolute value of $n$.

Real Numbers

This concept is often seen when $F$ is the set of real numbers $\R$.

Let $x, y \in \R$ be real numbers.

Then $x$ is an integral multiple of $y$ iff:

$\exists n \in \Z: x = n y$

Also see

Compare divisor, in which all the numbers involved are integers (or at least, elements of an integral domain).

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): Chapter $1 \ \S 2$