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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?

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  • $\begingroup$ What is $(X+Y)(X+Y)$? $\endgroup$ – Dietrich Burde Apr 21 at 12:47
  • $\begingroup$ $=0$ but I want a method or a procedure that I must use to find all of the zero divisor $\endgroup$ – Mary Maths Apr 21 at 12:48
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    $\begingroup$ "But I don't know how to start" - well, this was meant as a start for you. $\endgroup$ – Dietrich Burde Apr 21 at 12:55
  • $\begingroup$ I am not a native speaker that why I didn't express my self well. I change it now. My goal is to find all zero divisor not only one $\endgroup$ – Mary Maths Apr 21 at 13:10
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    $\begingroup$ Both rings are local and artinian, so the maximal ideal is the set of all zero-divisors. $\endgroup$ – user26857 Apr 21 at 22:41

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