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- W4296676400 abstract "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 Φ 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 sl2-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 Φ-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 sl2-triple and a is homogeneous of degree 2 both in the Z-grading of R and in the Φ-grading given by the eigenspaces of adh. Conversely, when there are enough invertible elements in Φ, we will show that if a nilpotent element a can be completed to an sl2-triple, then all powers ak are von Neumann regular." @default.
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- W4296676400 date "2023-01-01" @default.
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- W4296676400 title "Gradings induced by nilpotent elements" @default.
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- W4296676400 doi "https://doi.org/10.1016/j.laa.2022.09.017" @default.
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