I was reading Mumford's 'Red book on varieties and schemes', when I came across the following paragraph:
I am confused about meaning of the phrase "We let $k(X)$ be the zeroth graded piece of the localization of $R$ ..." and want to be sure I have understood it correctly.
Please correct me if I'm wrong: If $S$ denotes the set of all homogeneous elements of $R$, then $S$ is multiplicatively closed subset of $R$, and we give a gradation on $S^{-1}R$ by defining $\deg(f/g) := \deg(f)-\deg(g)$ for homogeneous elements $f,g\in R$. This defines a gradation on $S^{-1}R$, and $k(X)$ is the degree zero part of it.
But then what is the degree of the element $0$ in $S^{-1}R$? because I am getting trouble proving that $S^{-1}R$ is the direct sum of its graded pieces. I can see that they span the ring, but why is the sum direct?
Thanks in advance.