General Distributivity Theorem

Theorem
$$\left({R, \circ, \ast}\right)$$ be a ringoid.

Then for every sequence $$\left \langle {a_k} \right \rangle_{1 \le k \le n}$$ of terms of $$S$$, and for every $$b \in S$$:


 * $$\left({a_1 \circ \cdots \circ a_n}\right) \ast b = \left({a_1 \ast b}\right) \circ \cdots \circ \left({a_n \ast b}\right)$$
 * $$b \ast \left({a_1 \circ \cdots \circ a_n}\right) = \left({b \ast a_1}\right) \circ \cdots \circ \left({b \ast a_n}\right)$$

In the context of a ring, this can be translated into:

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

$$\forall n \in \mathbb{N}^*: \left({n \cdot x} \right) \circ y = n \cdot \left({x \circ y}\right) = x \circ \left({n \cdot y}\right)$$

Proof
This can be proved by induction.