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2 results

  • On Zero-divisors in Group Rings of Groups with Torsion

  • S. V. Ivanov, Roman Mikhailov
  • Journal:
    Canadian Mathematical Bulletin / Volume 57 / Issue 2 / 14 June 2014
    Published online by Cambridge University Press:
    20 November 2018, pp. 326-334
    Print publication:
    14 June 2014
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  • Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n\,\gg \,1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

  • ON THE ASPHERICITY OF LENGTH-6 RELATIVE PRESENTATIONS WITH TORSION-FREE COEFFICIENTS

  • Seong Kun Kim
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 51 / Issue 1 / February 2008
    Published online by Cambridge University Press:
    04 February 2008, pp. 201-214
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  • An interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.