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It is easy to see that there are no group divisors of length 2 in a group algebra of a torsion-free group. I saw somewhere mentioned that it is possible to do it for length 3. How?

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You should correct "group divisors" to "zero divisors" and maybe say that by "length" of an element you mean the cardinal of its support. –  YCor Jun 10 '12 at 11:04
    
Yves, this is how I read it))) –  Kate Juschenko Jun 10 '12 at 11:17
    
@Kate: I can read "zero divisors" in the title but "group divisors" in the text. In a first reading I couldn't understand the meaning of the question, and finally understood it after your answer. –  YCor Jun 10 '12 at 20:11

1 Answer 1

It is open for the group algebra over the field with $2$ elements, see

http://arxiv.org/abs/1202.6645 and http://arxiv.org/abs/1112.1790

and references there. I saw on the website of Mikhailov an unpublished paper, where he contributes this question to I. Rips.

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