A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as
$$A=\bigoplus_{n\geq 1} A_n$$
and the product satisfies $A_n \times A_m \rightarrow A_{nm}$.
A natural example of such a ring is the ring of Dirichlet polynomials:
$$A=\mathbf{C}[1^s, 2^s, 3^s, \dots]$$
with $\deg(n^s)=n$, $1^s=1$. More generally, if $M$ is a monoid equipped with a monoid morphism $\deg : M \to (\mathbf{N}_{>0}, \times)$, then the monoid algebra $\mathbf C[M]$ has a natural structure of multiplicatively graded algebra.
Are there other examples of such objects which arise naturally in mathematics?