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|Paper IPM / M / 14997||
Kaplanskyï¿½??s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field F, the group ring F[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in F[G] whose supports have size 3. For any
field F and all torsion-free groups G,we prove that if Î±Î² = 0 for some non-zero Î±, Î² ï¿½?? F[G] such that - supp(Î±) - = 3, then - supp(Î²) - ï¿½?ï¿½ 10. If F = F2 is the field with 2 elements, the latter result can be improved so that - supp(Î²) - ï¿½?ï¿½ 20. This
improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667ï¿½??693]. Concerning the unit conjecture, we prove that if Î±Î² = 1 for some Î±, Î² ï¿½?? F[G] such that - supp(Î±) - = 3, then - supp(Î²) - ï¿½?ï¿½ 9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326ï¿½??338] to arbitrary fields.
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