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| Paper IPM / M / 15369 |
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| Abstract: | |
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Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group
and F is a field, then the group ring F½G contains no zero divisor and
Kaplansky Unit Conjecture Q1 states that if G is a torsion-free group and F is a
field, then F½G contains no non-trivial units. The support of an element
a ¼ Px2G axx in F½G, denoted by suppða�?, is the set fx 2 Gjax 6¼ 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 F½G for a possible torsion-free group G and an arbitrary field
F such that jsuppða�?j ¼ 4 and ab ¼ 0, then jsuppðb�?j 7. In [J. Group
Theory, 16 ð2013�?; no. 5, 667�??693], it is proved that if F ¼ F2 is the field
with two elements, G is a torsion-free group and a; b 2 F2½G n f0g such
that jsuppða�?j ¼ 4 and ab ¼ 0, then jsuppðb�?j 8. We improve the latter
result to jsuppðb�?j 9. Also, concerning the Unit Conjecture, we prove
that if ab ¼ 1 for some a; b 2 F½G and jsuppða�?j ¼ 4, then jsuppðb�?j 6.
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