[Math] Zero divisor conjecture for finite fields

gr.group-theoryopen-problemsra.rings-and-algebras

I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let me ask if the zero conjecture is known for finite fields instead of $\mathbb{C}$. More precisely:

  • Let $G$ be a torsion-free group and $K$ be a finite field. Is it true that the group ring $K[G]$ has no non-trivial zero divisors?

In particular:

  • Let $K$ be the field with two elements, $G$ be a torsion free group and let $rank(a)$ be the smallest number of elements in the expression of $a$ in the sum $a=\sum g_i$, $g_i\in G$. Is there a constant $R>0$ such that we know for sure that $rank(a)>R$ for every zero-divisor $a\in K[G]$?

Here is related question:

Group ring and left zero divisor.

Best Answer

@Kate: I believe that the answer to the first question is unknown and the question is considered as complicated for the field $\mathbb{F}_2$ as for $\mathbb{C}$. I do not know any reduction from the case of one field to the case of another field, though. There were several attempts to disprove Kaplansky conjecture for $\mathbb{F}_2$ but there are no promising ideas.

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