Kaplansky's zero divisor and unit conjectures on elements with supports of size 3

2016/12/03 by Abdollahi, Alireza, Taheri, Zahra
#16S34 #20C07 #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1612.00934

Abstract

Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field \mathbbF, the group ring \mathbbF[G] has no zero divisors (has no unit with support of size greater than 1). In this paper, we study possible zero divisors and units in \mathbbF[G] whose supports have size 3. For any field \mathbbF and all torsion-free groups G, we prove that if αβ=0 for some non-zero α, β∈ \mathbbF[G] such that |supp(α)|=3, then |supp(β)|≥ 10. If \mathbbF=\mathbbF₂ is the field with 2 elements, the latter result can be improved so that |supp(β)|≥ 20. This improves a result in [J. Group Theory, 16 (2013), no. 5, 667-693]. Concerning the unit conjecture, we prove that if αβ=1 for some α, β∈ \mathbbF[G] such that |supp(α)|=3, then |supp(β)|≥ 9. The latter improves a part of a result in [Exp. Math., 24 (2015), 326-338] to arbitrary fields.

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