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 commutative semiring.

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

Consider the natural numbers $\N$ defined as the naturally ordered semigroup $\left({S, \circ, \preceq}\right)$.

Let $\times$ be the multiplication operation on $\left({S, \circ, \preceq}\right)$.

Then we have:


 * $(1): \quad$ From Multiplication in Naturally Ordered Semigroup is Closed, $\times$ is closed on $S$


 * $(2): \quad$ From Multiplication in Naturally Ordered Semigroup is Associative, $\times$ is associative on $S$


 * $(3): \quad$ From Multiplication in Naturally Ordered Semigroup is Commutative, $\times$ is commutative on $S$


 * $(4): \quad$ From Multiplication in Naturally Ordered Semigroup is Distributive, $\times$ is distributive over $\circ$.

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

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

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