Zero divisors and units with small supports in group algebras of torsion-free groups. Issue 2 (1st February 2018)

• Record Type: Journal Article
• Authors: Abdollahi, Alireza; Taheri, Zahra
• Is Part Of: Communications in algebra. Volume 46:Issue 2(2018)
• Page Start: 887
• Page End: 925
• Publication Date: 2018-02-01
• Subjects: Divisor graph -- group ring -- Kaplansky's unit conjecture -- Kaplansky's zero divisor conjecture -- torsion-free group -- zero divisor -- zero unit graph; 20C07 -- 16S34; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1344688
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Kaplansky'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 αβ = 0 for some non-zero α, β ∈ 𝔽 [ G ] such that | supp ( α )| = 3, then | supp ( β )|≥10. If 𝔽 = 𝔽 2 is the field with 2 elements, the latter result can be improved so that | supp ( β )|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α, β ∈ 𝔽 [ G ] such that | supp ( α )| = 3, then | supp ( β )|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields.