Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$
Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0.
Let $f$ be a polynomial which is zero on the zero-set of the $p$’s.
Conjecture: $f$ must be in the ideal generated by the $p$’s.
Does this conjecture hold? If so it would be a Nullstellensatz of the form $I=J(V(I))$ for these ideals.
I find this easier to reason about with a contrapositive: Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0. Then for any polynomial $f$ which is not in their ideal, there is some $s=(s_1,\ldots,s_n)\in R^n$ with all $p_i(s)=0$ and $f(s)\neq 0$.
A first example is $p=x^{k+1}$. Then $f=x^k$ is not in the ideal of $p$, and $s=t_1+\cdots +t_k$ satisfies $p(s)=0$ (because each $t_i^2$ is zero) and $f(s)=k!\prod t_i \neq 0$.
As another example, with $p_1=x^3$, $p_2=y^3$, $p_3=x^2+y^2$, and $f=x^2-y^2$, we can take $s=(t_1+t_2,t_1-t_2)$.
I once wrote down a proof of this conjecture for a large class of examples at this level of difficulty. For more complicated polynomials like $x^2+xy+y^3$ I have neither a proof nor a counterexample.
I make the conjecture because of an analogy with Smooth Infinitesimal Analysis, which I’ve explained in another answer. Specifically, this conjecture is analogous to a corrected version of the uniqueness in the generalized Kock-Lawvere axiom from Models For Smooth Infinitesimal Analysis by Ieke Moerdijk and Gonzalo Reyes; the correction, which I found by exploring this analogy, is that the ideal in the book’s statement should include the polynomials $x_i^{k+1}$.
There might be a slick topos-theoretic proof, or there might be a helpful combinatorial construction; I’d welcome any results.