In Thm 3.4 in Hartshorne and comments above it, it says: "Similarly, if $f \in S$ is a homogeneous element, we denote by $S_{(f)}$ the subring of elements of degree $0$ in the localized ring $S_{f}$."
Now, in this case, what is the multiplicative set of the localized ring $S_{f}$; is it just $\{f\}$ or the ideal generated by f i.e. $(f)$.
It's the set of powers of $f$, not the ideal.