This is generalization of the univariate case and also related to open problem.

Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_k(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with integer coefficients.

Let $K=(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/<f_1(x_i),f_2(x_i)...f_k(x_i)>$.

Let $S$ be the set of degree 2 nilpotent elements of $K$ of degree at most $B$:

$$S=\{g : g \in K, g^2=0,\deg(g) \le B \}$$

I am trying to find large subset of $S'$ of $S$ such that $\prod S' \ne 0$.

Q1 How large can $S'$ be?

Lower bound for $|S'|$ is $m+1$ by setting $k=m$ and $n=n_0^2$ and $f_i=x_i^2$ and $S'=\{n_0,x_1,x_2,...x_m\}$.

Disadvantage of this construction is that working in this quotient is not efficient because of exponential growth of monomials.