We 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 nonzero power xn is regular von Neumann, i.e., there exists another element y∈R such that xnyxn=xn. 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 Qms(R) of R we still obtain inner derivations with the same indices of nilpotence on Qms(R) and on the skew-symmetric elements Skew(Qms(R),∗) of Qms(R), but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of Qms(R) and of Skew(Qms(R),∗). We will review the Jordan canonical form of nilpotent last-regular elements and show how to get gradings in associative algebras (with and without involution) when they have such elements.
This video is part of the European Non-Associative Algebra Seminar series.
