Definition:Integral Multiple/Rings and Fields
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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:
- $n \cdot a := \begin {cases}
0_F & : n = 0 \\ \paren {\paren {n - 1} \cdot a} + a & : n > 1 \\ \size n \cdot \paren {-a} & : n < 0 \\ \end {cases}$ where $\size n$ is the absolute value of $n$.
Using sum notation:
- $\ds n \cdot a := \sum_{j \mathop = 1}^n a$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties