Natural Numbers form Commutative Semiring

Theorem
The semiring of natural numbers $\struct {\N, +, \times}$ forms a commutative semiring.

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

Then we have:


 * $(1): \quad$ $\times$ is closed


 * $(2): \quad$ Natural Number Multiplication is Associative


 * $(3): \quad$ Natural Number Multiplication is Commutative


 * $(4): \quad$ Natural Number Multiplication Distributes over Addition

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

So by definition, $\struct {\N, \times}$ is a commutative semigroup.

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