Multiple of Ring Product

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 $\text {(1)}: \quad$ $\displaystyle$ $=$ $\displaystyle \sum_{j \mathop = 1}^n \left({x \circ y} \right)$ 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

$\blacksquare$