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I think (my maths could be wrong up to this point) that I am working with the ring:

\begin{equation} \mathbb{C}[x, y, z]/(x^3 − y^3 − z^3,x^3,x^2z,xz^2,z^3)=\mathbb{C}[x, y, z]/(− y^3,x^3,x^2z,xz^2,z^3) \end{equation}

and I want to find a zero divisor of this ring which is not nilpotent. I have no idea how to do this, any suggestions? Of course, my ring could be wrong due to a wrong choice of Prime Ideal, so if there is no zero divisors that aren't nilpotent then that's what's wrong.

Edit1: Thanks to hints I now know this is not the correct ring. I started with the ring $R=\mathbb{C}[x,y,z]/(x^3-y^3-z^3)$ and used the prime ideal $p=(x,z)$. Taking $p^3$ and forming the quotient ring $R/p^3$ I got to this point.

I want to find a prime ideal where $p^3$ is not a primary ideal. Any suggestions how to find such a prime ideal?

Edit2: I have now posted Edit1 as a new question. Thanks for the feedback everyone

MeBadMaths
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  • Hint: in this ring $R$, $(x,y,z)$ is a nilpotent maximal ideal and there’s an easy map of rings $R/(x,y,z) \cong \mathbb{C} \rightarrow R$. It follows that any element outside $(x,y,z)$ can be written as $\lambda + \nu$ with $\nu$ nilpotent and $\lambda \in \mathbb{C}^*$. Show that such an element must always be invertible (hence not a divisor of zero). – Aphelli Jan 12 '21 at 14:19
  • Thanks for your reply, much appreciated!

    However, I struggle to understand isomorphic/mapping concepts (not great for this commutative algebra module). So a few questions:

    1. What do you mean by "an easy map"?
    2. Any element outside of $(x,y,z)$, is that to say any point that isn't $(x,y,z)$, or is that the Ideal generated by ${x,y,z}$?
    3. If I am showing that it is invertible, and therefore not a zero devisor, is that not what I am trying to find?

    That hint seems to read that I have a nilpotent Max ideal, and I want to show it's not a 0-divisor. Is that the same as what I asked for?

     – MeBadMaths Jan 12 '21 at 17:11
  • I didn’t know how to name that map, but it really is the inclusion $\mathbb{C} \subset R$. 2) I mean an element of $R$ which isn’t in the ideal $(x,y,z)$. 3) I suggest that you prove that the elements outside that maximal (nilpotent) ideal are invertible. Thus the ring can be partitioned between nilpotent elements and invertible elements (thus no not-nilpotent element is a zero-divisor).
    • – Aphelli Jan 12 '21 at 18:51
  • Thanks Mindlack, I can see now why this Ring isn't what I need. THe Ring was made with the intent of looking for a quoient ring who has a prime ideal which isn't a primary ring at some power. I thought my choice of prime ring might have been sufficient, but (as said in the question) my choice was wrong.

    Is there any indication to find such a prime ring? Or is it just luck to find it? Thank you for your hint, again it's appreciated

     – MeBadMaths Jan 14 '21 at 15:48
  • Extra: Original ring was $R=\mathbb{C}[x,y,z]/(x^3-y^3-z^3)$. The prime Ideal I tried was $p=(x,z)$ and looked at $R/p^3$. Clearly, that was wrong – MeBadMaths Jan 14 '21 at 15:59
  • Use this example and try to show that all powers (from $2$ on) of that prime ideal are not primary. – user26857 Jan 14 '21 at 21:43
  • I've seen that example a good few times, but found it to be just different enough to shed minimal help. I know that I need to eliminate the cube powers in my example, but the wiki example has a fortunate xy term that helps out. Also, thank you for the recommendation, I have now posted my edit as a new question. Thanks! – MeBadMaths Jan 15 '21 at 12:15