Are all graded rings actually graded algebras?

Stream: mathlib4

[Kenny Lau]

So I don't mean it in the form of "they're all graded ℤ-algebra" or "(𝒜 0)-algebra", my question is:

Is there an organic example of [GradedRing 𝒜] where 𝒜 is not syntactically 𝒜 : ι → Submodule R A?

[Eric Wieser]

You could have one for FreeRing I assume

[Kenny Lau]

then 𝒜 : ι → AddSubmonoid (FreeRing α)?

[Eric Wieser]

AddSubgroup