4
$\begingroup$

This is related to an open problem.

Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.

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

$$ S=\{g : g \in K, g^2=0\} $$.

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

Numerical experiments suggest $|S'|=1$.

Q1 How large can $S'$ be?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ You can definitely have $|S'|=2$: Take $\mathbb{Z}/\langle p^2, x^2 \rangle$, and $S' = \{ x, p \}$. I suspect you can't do better than this, but no proof yet. $\endgroup$
    – David E Speyer
    Commented Dec 28, 2023 at 17:03
  • $\begingroup$ @DavidESpeyer Thanks. I found another |S'|=2 for $n=3^2$ and $f=((x-1)(x-2))^2$, I believe squares help. Computer search is not very fast. $\endgroup$
    – joro
    Commented Dec 28, 2023 at 17:37

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .