Cancellable Semiring with Unity is Additive Semiring

Theorem
In a cancellable semiring $\left({S, *, \circ}\right)$ with unity $1_R$, 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 for all $a, b, c, d \in \left({S, *, \circ}\right)$, it is true in particular if $c = d = 1_R$.

Thus it is clear that $b * a = a * b$, which is exactly to say that $*$ is commutative.