Didn't get a single comment in over a day at math.SE, so maybe the question is more appropriate here.

I'm looking for a reference to a generalization of Hilbert's Nullstellensatz to the multiprojective setting. This would appear to be a rather basic fact but I'm having trouble finding a statement online or in the literature.

Consider the product $$\mathbb P=\mathbb P(\mathbb C^{n_1})\times\dots\times \mathbb P(\mathbb C^{n_k}).$$ Let $R$ be the ring of polynomials in variables $X_i^j$ with $j\in[1,k]$ and $i\in[1,n_j]$. Let $I\subset R$ be a *multihomogeneous* ideal, meaning that $I$ is homogeneous with respect to degree in the variables $X_1^j,\dots,X_{n_j}^j$ for every $j$. Then $I$ is seen to define a subvariety $V(I)\subset\mathbb P$ of points in which all $p\in I$ vanish. **Question:** where in the literature can I find a necessary and sufficient condition for $I$ to contain *all* polynomials vanishing on $V(I)$? (Preferably in a standard reference such as Hartshorne.)

I suspect that the condition is that $I$ is (a) multihomogeneous, (b) radical and (c) saturated with respect to the irrelevant ideals $\langle X_1^j,\dots,X_{n_j}^j\rangle$ for all $j$. I think I know how this can be proved but I'm looking for a precise reference.