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 $+$.