Natural Numbers form Commutative Semiring

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

Proof
The algebraic structure $$\left({\mathbb{N}, +}\right)$$ is a commutative monoid from Natural Numbers under Addition is 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 $$S$$;
 * distributive over $$\circ$$.

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

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

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