I don't know how to find all zero divisors for polynomials in several variables. For example:
$\mathbb{Z}_2[X,Y]/(X^2,XY,Y^2)\quad $ or $\quad \mathbb{Z}_4[X,Y]/(X^2,Y^2-XY)$
Can we to proceed like in $\mathbb{Z}_2[X]$ over a non irreducible polynomial by using euclidian divison, and find all zero divisors for this ring?