Multiple of Ring Product

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: