Ring of Arithmetic Functions is Ring with Unity

Theorem
Let $\mathcal A$ be the set of all arithmetic functions.

Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions.

The ring of arithmetic functions $\struct {\mathcal A, +, *}$ is a commutative ring with unity.

Proof
By Structure Induced by Abelian Group Operation is Abelian Group, $\struct {\mathcal A, +}$ is an abelian group.

By Properties of Dirichlet Convolution, $*$ is commutative, associative and has unity.

Therefore $\struct {\mathcal A, +, *}$ is a commutative ring with unity.