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:

$(1): \quad$ Natural Number Multiplication 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 $+$.

$\blacksquare$