# Natural Numbers form Commutative Semiring

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## 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

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 $+$.

$\blacksquare$