Gradings induced by nilpotent elements

An element a is nilpotent last-regular if it is nilpotent and its last nonzero power is von Neumann regular. In this paper we show that any nilpotent last-regular element a in an associative algebra R over a ring of scalars phi gives rise to a complete system of orthogonal idempotents that induces a finite Z-grading on R; we also show that such element gives rise to an s[2-triple in R with semisimple adjoint map adh, and that the grading of R with respect to the complete system of orthogonal idempotents is a refinement of the phi-grading induced by the eigenspaces of adh. These results can be adapted to nilpotent elements a with all their powers von Neumann regular, in which case the element a can be completed to an s[2-triple and a is homogeneous of degree 2 both in the Z-grading of R and in the phi-grading given by the eigenspaces of adh.