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SUMMARY:Esther García González (King Juan Carlos University\, Spain)
DTSTART:20230626T150000Z
DTEND:20230626T160000Z
DTSTAMP:20260423T052548Z
UID:ENAAS/24
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ENAAS/24/">N
 ilpotent last-regular elements</a>\nby Esther García González (King Juan
  Carlos University\, Spain) as part of European Non-Associative Algebra Se
 minar\n\n\nAbstract\nWe say that an element $x$ in a ring $R$ is nilpotent
  last-regular if it is nilpotent of certain index $n+1$ and its last nonze
 ro power $x^n$ is regular von Neumann\, i.e.\, there exists another elemen
 t $y\\in R$ such that $x^nyx^n=x^n$. This type of elements naturally arise
  when studying certain inner derivations in the Lie algebra $\\Skew(R\,*)$
  of a ring $R$ with involution $*$ whose indices of nilpotence differ when
  considering them acting as derivations on $\\Skew(R\,*)$ and on the whole
  $R$. When moving to the symmetric Martindale ring of quotients $Q^s_m(R)$
  of $R$ we still obtain inner derivations with the same indices of nilpote
 nce on $Q^s_m(R)$ and on the skew-symmetric elements $\\Skew(Q^s_m(R)\,*)$
  of $Q^s_m(R)$\, but with the extra condition of being generated by a nilp
 otent last-regular element. This condition strongly determines the structu
 re of $Q^s_m(R)$ and of $\\Skew(Q^s_m(R)\,*)$. \nWe will review the Jordan
  canonical form of nilpotent last-regular elements and show how to get gra
 dings in associative algebras (with and without involution) when they have
  such elements.\n
LOCATION:https://researchseminars.org/talk/ENAAS/24/
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