what's the meaning of $B_m/f^n$?

Question

In Page 82 of Qing Liu's book "Algebraic Geometry and Arithmetic Curves", $B$ is a graded ring and $f\in B_+$ is a homogeneous element, it says $B_{(f)}$ is a direct factor of $B_{f}=B_{(f)}\oplus(\oplus_{m\neq n \operatorname{deg}f}B_m/f^n)$, what's the meaning of $B_m/f^n$?

Answer 1

It means elements of $B$ of degree $m$ divided by $f^n$. For example if $B=k[x]$ then an element of $B_m/f^n$ looks like $\frac{cx^m}{f^n}$ where $c\in k$. Just think of that direct sum decomposition as chopping up $B_f$ into the degree $0$ parts, which is $B_{(f)}$, and everything else.