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Paper IPM / M / 15369  


Abstract:  
Kaplansky Zero Divisor Conjecture states that if G is a torsionfree 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 torsionfree group and F is a
field, then FÂ½G contains no nontrivial 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 torsionfree groups. We prove that if a, b are nonzero
elements in FÂ½G for a possible torsionfree 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 torsionfree 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.
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