# Definition:Integral Multiple/Rings and Fields

## Definition

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:

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

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

Using sum notation:

$\displaystyle n \cdot a := \sum_{j \mathop = 1}^n a$