Multiple of Ring Product
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Let $x, y \in \struct {R, +, \circ}$.
Then:
- $\forall n \in \Z_{> 0}: \paren {n \cdot x} \circ y = n \cdot \paren {x \circ y} = x \circ \paren {n \cdot y}$
where $n \cdot x$ denotes the $n$th multiple of $x$.
Proof
By definition:
- $\ds n \cdot x := \sum_{j \mathop = 1}^n x$
Thus:
\(\ds \paren {n \cdot x} \circ y\) | \(=\) | \(\ds \paren {\sum_{j \mathop = 1}^n x} \circ y\) | Definition of Integral Multiple | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^n \paren {x \circ y}\) | General Distributivity Theorem | ||||||||||
\(\ds \) | \(=\) | \(\ds n \cdot \paren {x \circ y}\) | Definition of Integral Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \paren {\sum_{j \mathop = 1}^n y}\) | General Distributivity Theorem from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \paren {n \cdot y}\) | Definition of Integral Multiple |
$\blacksquare$