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Paper   IPM / M / 15369
 School of Mathematics Title: Zero divisor and unit elements with supports of size 4 in group algebras of torsion-free groups Author(s): Alireza Abdollahi (Joint with F. Jafari) Status: To Appear Journal: Comm. Algebra Supported by: IPM
Abstract:
Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and F is a field, then the group ring FÂ½G contains no zero divisor and Kaplansky Unit Conjecture Q1 states that if G is a torsion-free group and F is a field, then FÂ½G contains no non-trivial units. The support of an element a Â¼ Px2G axx in FÂ½G, denoted by suppÃ°aï¿½?, is the set fx 2 Gjax 6Â¼ 0g. 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 a, b are non-zero elements in FÂ½G for a possible torsion-free group G and an arbitrary field F such that jsuppÃ°aï¿½?j Â¼ 4 and ab Â¼ 0, then jsuppÃ°bï¿½?j  7. In [J. Group Theory, 16 Ã°2013ï¿½?; no. 5, 667ï¿½??693], it is proved that if F Â¼ F2 is the field with two elements, G is a torsion-free group and a; b 2 F2Â½G n f0g such that jsuppÃ°aï¿½?j Â¼ 4 and ab Â¼ 0, then jsuppÃ°bï¿½?j  8. We improve the latter result to jsuppÃ°bï¿½?j  9. Also, concerning the Unit Conjecture, we prove that if ab Â¼ 1 for some a; b 2 FÂ½G and jsuppÃ°aï¿½?j Â¼ 4, then jsuppÃ°bï¿½?j  6.