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Let $S$ be a graded ring and $f$ is an element of degree $d>0$. Then $S_{(f)}$ is defined as the subring of elements of degree $0$ in the localized ring $S_{f}$ (Hartshorne, pp. 77). But the definition seems to have a problem.

Suppose that the graded ring $S$ is not a domain, and $fy=0$ for some non-zero element $y$. Then $\frac{x+y}{f^n}=\frac{x}{f^n}\in S_f$ for each $x\in S$ and $n\in\mathbb N$. When $\text{deg}(x)=nd\ne \text{deg}(y)$, is the above element included in $S_{(f)}$?

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Yes. For an element of the localization to be homogeneous of some degree, all that is required is that it can be represented by a homogeneous fraction of that degree. It is not necessary that every representation as a fraction is homogeneous of that degree.

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