I am a beginner studying algebraic geometry on the level of Hartshorne. By Chapter II Exercise 2.14(c), the projective spectra $\mathrm{Proj}\,S$ can be isomorphic (as locally ringed spaces) to $\mathrm{Proj}\,T$ for non-isomorphic graded rings $S$ and $T$, as long as $S_d\cong T_d$ for all $d\ge d_0$, where $d_0$ is an integer. This seems a little surprising to me because it is very different from affine spectra.
I am curious if we can uniquely determine the rings $S$ or $T$, if we put restriction on the degrees of the generators. More precisely, let $R:= k[x_0,\dots,x_n]$ denote a polynomial ring over a field $k$, with ideals $I,J\subseteq R$. If we assume that the ideals $I,J$ are both generated by homogeneous polynomials with a fixed degree (e.g., quadratic forms), do we necessarily have that $\mathrm{Proj}\,(R/I)\cong\mathrm{Proj}\,(R/J)$ implies $I\cong J$ (i.e., there exists an automorphism of $R$ under which $I$ and $J$ can be identified)? Any comment or reference would be greatly appreciated!