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.

Then $(\mathcal A, +, *)$ is a commutative ring with unity.

Proof
By Induced Group, $(\mathcal A, +)$ is an abelian group.

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

Therefore, $(\mathcal A, +, *)$ is a commutative ring with unity.