Integer Multiples form Commutative Ring

Theorem
Let $$n \Z$$ be the set of integer multiples of $$n$$.

Then $$\left({n \Z, +, \times}\right)$$ is a commutative ring.

Proof
From Additive Group of Integer Multiples, $$\left({n \Z, +}\right)$$ is a cyclic group, therefore abelian from Cyclic Group is Abelian.

From Integer Multiples Closed under Multiplication and Integer Multiplication is Associative, we have that $$\left({n \Z, \times}\right)$$ is a semigroup.

From Integer Multiplication Distributes over Addition it follows that $$\left({n \Z, +, \times}\right)$$ is a ring.

Finally, from Integer Multiplication is Commutative we have that $$\left({n \Z, +, \times}\right)$$ is a commutative ring.