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