Which ideals on $S \otimes_R T$ correspond to closed subschemes $Proj(S) \times_{Spec(R)} Proj(T)$?

Let $$S$$ and $$T$$ be graded rings, finitely generated in degree 1 over degree 0. Suppose furthermore that $$S_0$$ and $$T_0$$ are isomorphic (call it $$R$$). I want to determine which kinds of ideal on $$S \otimes_R T$$ correspond to closed subschemes of $$Proj(S) \times_{Spec(R)} Proj(T)$$, and how to determine if two ideals give the same closed subscheme.

I know this should be possible, since there is the Segre embedding $$\mathbb{P}^m_R \times_R \mathbb{P}^n_R \rightarrow \mathbb{P}^N_R$$, so any closed subscheme $$Z \rightarrow Proj(S) \times_{Spec(R)} Proj(T)$$ is also a closed subscheme of $$\mathbb{P}^N_R$$.

If $$F$$ is a quasicoherent sheaf of ideals on $$Proj(S) \times_{Spec(R)} Proj(T)$$, then on $$U = D(s) \times D(t)$$ (where $$s \in S, t \in T$$ are homogeneous of degree 1), $$F(U)$$ is an ideal of $$S_{s : 0} \otimes_R T_{t : 0}$$, and this ideal must agree on overlaps. Thus, I think that the ideals of $$S \otimes_R T$$ corresponding to closed subschemes of $$Proj(S) \times_{Spec(R)} Proj(T)$$ should be the "doubly homogeneous ideals" : if $$x = \sum_{i, j} s_i \otimes t_j \in I$$, where each $$s_i \in S, t_j \in T$$ are homogeneous of degrees $$i$$ and $$j$$, then $$s_i \otimes t_i \in I$$. But how do I prove this?

Let me assume that either $$\mathrm{Proj}(S)$$ or $$\mathrm{Proj}(T)$$ is flat over $$R$$ (more generally, it is enough to assume that they are $$\mathrm{Tor}$$-independent). Then the Künneth formula (applied to the Segre embedding) implies that $$\mathrm{Proj}(S) \times_{\mathrm{Spec}(R)} \mathrm{Proj}(T) \cong \mathrm{Proj} \left( \bigoplus_{k=0}^\infty S_k \otimes_{R} T_k \right).$$ The graded algebra in the right side is generated by its first component, so you can associate closed subschemes to ideals in it in the usual way.