Zero divisor and unit elements with supports of size 4 in group algebras of torsion-free groups. Issue 1 (2nd January 2019)

• Record Type: Journal Article
• Authors: Abdollahi, Alireza; Jafari, Fatemeh
• Is Part Of: Communications in algebra. Volume 47:Issue 1(2019)
• Page Start: 424
• Page End: 449
• Publication Date: 2019-01-02
• Subjects: Group ring -- Kaplansky unit conjecture -- Kaplansky zero divisor conjecture -- torsion-free group; 20C07 -- 16S34; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2018.1477949
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and F is a field, then the group ring F [ G ] contains no zero divisor and Kaplansky Unit Conjecture states that if G is a torsion-free group and F is a field, then F [ G ] contains no non-trivial units. The support of an element α = ∑ x ∈ G α x x in F [ G ], denoted by supp ( α ), is the set { x ∈ G | α x ≠ 0 } . In this paper, we study possible zero divisors and units with supports of size 4 in group algebras of torsion-free groups. We prove that if α, β are non-zero elements in F [ G ] for a possible torsion-free group G and an arbitrary field F such that | supp ( α ) | = 4 and α β = 0, then | supp ( β ) | ≥ 7 . In [J. Group Theory, 16 ( 2013 ), no. 5, 667–693], it is proved that if F = F 2 is the field with two elements, G is a torsion-free group and α, β ∈ F 2 [ G ] ∖ { 0 } such that | supp ( α ) | = 4 and α β = 0, then | supp ( β ) | ≥ 8 . We improve the latter result to | supp ( β ) | ≥ 9 . Also, concerning the Unit Conjecture, we prove that if a b = 1 for some a, b ∈ F [ G ] and | supp ( a ) | = 4, then | supp ( b ) | ≥ 6 .