Cancellable Semiring with Unity is Additive Semiring

Theorem
In a cancellable semiring $$\left({S, *, \circ}\right)$$, the distributand $$*$$ is always commutative.

Proof
Let $$\left({S, *, \circ}\right)$$ be a semiring, all of whose elements of $$S$$ are cancellable for $$*$$.

We expand the expression $$\left({a * b}\right) \circ \left({c * d}\right)$$ using the distributive law in two ways:

So, by the fact that all elements of $$\left({S, *}\right)$$ are cancellable (and thus are $$a \circ c$$ and $$b \circ d$$), we have:

As this is true $$\forall a, b, c, d \in \left({S, *, \circ}\right)$$, it follows that for the distributive law to work, then $$*$$ must be commutative. \end {proof}