Proj
0.1 Homogeneous Submonoids
Let \(A\) be a graded ring.
We say \(S\) is a homogeneous submonoid of \(A\) if \(S\) is a submonoid of \(A\) and for every \(s \in S\), \(s\) is a homogeneous element of \(A\).
If \(A\) is graded by \(M\) where \(M\) is an abelian group written additively, then \(A_{S}\) is a graded ring by \(M\) as well.
Let \(S\) be a homogeneous submonoid of \(A\), we define \(\deg (S)\) to be the submonoid of \(M\) to be
\[ \{ i | \exists x \in S, x \in A_i\} . \]