I am genuinely struggling with this concept, partly because of the language of graded rings.
So, I will ask a simple question. If polynomial rings $\mathbb{k}[x_0,x_1]$ and $\mathbb{k}[y_0,y_1]$ with associated weighted systems $S(1,1)$ and $S(2,2)$. Can I say that in the weighted systems there is an isomorphism of graded rings since I can define a map $\phi: x_0,x_1 \mapsto y_0,y_1$ which will preserve homogeneity?
This induces an isomorphism of weighted projective spaces $\mathbb{P}(1,1)$ and $\mathbb{P}(2,2)$.
Thus allowing us to saying that $\mathbb{P}(a_0,...,a_n)$ isomorphic to $\mathbb{P}(qa_0,...,qa_n)$.
I am asking this silly question because some texts ask for degrees to be preserved for an isomorphism graded rings