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

$\blacksquare$