Multiple of Ring Product

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Theorem

Let $\left({R, +, \circ}\right)$ be a ring.

Let $x, y \in \left({R, +, \circ}\right)$.


Then:

$\forall n \in \Z_{> 0}: \left({n \cdot x} \right) \circ y = n \cdot \left({x \circ y}\right) = x \circ \left({n \cdot y}\right)$

where $n \cdot x$ denotes the $n$th multiple of $x$.


Proof

By definition:

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

Thus:

\(\displaystyle \left({n \cdot x} \right) \circ y\) \(=\) \(\displaystyle \left({\sum_{j \mathop = 1}^n x} \right) \circ y\) by definition of integral multiple
\((1):\quad\) \(\displaystyle \) \(=\) \(\displaystyle \sum_{j \mathop = 1}^n \left({x \circ y} \right)\) General Distributivity Theorem
\(\displaystyle \) \(=\) \(\displaystyle n \cdot \left({x \circ y}\right)\) by definition of integral multiple
\(\displaystyle \) \(=\) \(\displaystyle x \circ \left({\sum_{j \mathop = 1}^n y} \right)\) General Distributivity Theorem from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle x \circ \left({n \cdot y}\right)\) by definition of integral multiple

$\blacksquare$