Questions on (subring/ submodule) of a graded (ring/ module)

Question

I have a question which seems a bit silly...

If we have $R$ a graded ring, does it follow that every subring of $R$ is also graded?

Because I have a problem here as such: I have a graded ring $R$ and a subring $A$. Does it follow that $R$ is a graded $A$-module?

Thanks for your help!

Answer 1

I think you need to have that $A$ is a graded subring to have a canonical $A$-module structure on $R$. (I'm supposing you want to have a grading on A which is inherited by that of $R$, otherwise you just trivially grade $A$ and you get it).