\begin{definition}[Definition:Integral Multiple/Rings and Fields]
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:
:$\ds n \cdot a := \sum_{j \mathop = 1}^n a$
\end{definition}