Natural Numbers form Commutative Semiring

Theorem
The algebraic structure $$\left({\N, +, \times}\right)$$ consisting of the set of natural numbers $$\N$$ under addition $$+$$ and multiplication $$\times$$ forms a semiring in which $$\times$$ is commutative.

Proof
From Natural Numbers are a Naturally Ordered Semigroup, the algebraic structure $$\left ({\N, +}\right)$$ is a naturally ordered semigroup.

The algebraic structure $$\left({\N, +}\right)$$ is a commutative monoid from Natural Numbers under Addition is Commutative Monoid.

From Naturally Ordered Semigroup with Product, the operation $$\times$$ on a naturally ordered semigroup $$\left({S, \circ, \preceq}\right)$$ is:


 * closed on $$S$$


 * associative on $$S$$


 * commutative on $$S$$


 * distributive over $$\circ$$.

Thus $$\left({\N, \times}\right)$$ forms an algebraic structure which is closed such that $$\times$$ is associative.

So by definition, $$\left({\N, \times}\right)$$ is a semigroup.

The result follows from definition of semiring and the distributivity of $$\times$$ over $$+$$.