I'm trying to answer the following questions:
Let $A$ be a (not necessarily commutative) $\mathbb{Z}$-graded ring and $S$ a multiplicative subset of $A$ such that $AS^{-1}$ exists. Is $AS^{-1}$ a $\mathbb{Z}$-graded ring?
And the same exercise with "filtrated" in the place of "graded".
I've tried to show that this is true, but with no success: Let $\{A_p\}_{p\in \mathbb{Z}}$ be a graduation for $A$. Define $AS^{-1}_p=\{\frac{a}{s}\in AS^{-1}\, |\, a\in A_p\}$ and consider $\{AS^{-1}_p\}_{p\in \mathbb{Z}}$. I'd like to see that this is a graduation for $AS^{-1}$, but i can't even show that those are additive subgroups of $AS^{-1}$. I would appreciate any orientations for both exercises.