Natural Numbers form Commutative Semiring

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Theorem

The semiring of natural numbers $\left({\N, +, \times}\right)$ forms a commutative semiring.


Proof

The algebraic structure $\left({\N, +}\right)$ 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 $\left({\N, \times}\right)$ 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 $+$.

$\blacksquare$