Definition:Integral Multiple/Rings and Fields

Definition

Let $\struct {F, +, \times}$ be a ring or a field whose zero is $0_F$.

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:

$n \cdot a := \begin {cases} 0_F & : n = 0 \\ \paren {\paren {n - 1} \cdot a} + a & : n > 1 \\ \size n \cdot \paren {-a} & : n < 0 \\ \end {cases}$

where $\size n$ is the absolute value of $n$.

Using sum notation:

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