Graded rings and maximal ideals of a local ring

Question

Let $S$ be a graded ring,$p$ a homogeneous prime ideal in $S$, and $S_{(p)}$ the subring of elements of degree 0 in the localization of $S$ with respect to the multiplicative subset $T$ consisting of homogeneous elements of $S$ not in $p.$

1. How do the elements of $S_{(p)}$ look like ? In the case $S$ denotes the polynomial ring in $n$ indeterminates, are these elements the constant polynomials ?
2. How one can determine the maximal ideal of $S_{(p)}$ ?

Thanks.

Answer 1

1. The elements are quotients $f/g$ where $f$ and $g$ are polynomials of same degree and $g$ is not contained in p.

2. The maximal ideal consists of quotiensts $f/g$ where $f$ is contained in $p$. All other elements are clearly invertible.