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On one-sided Lie nilpotent ideals of associative rings
We prove that a Lie nilpotent one-sided ideal of an associative ring is
contained in a Lie solvable two-sided ideal of . An estimation of derived
length of such Lie solvable ideal is obtained depending on the class of Lie
nilpotency of the Lie nilpotent one-sided ideal of One-sided Lie nilpotent
ideals contained in ideals generated by commutators of the form are also studied.Comment: 5 page
Commuting varieties of -tuples over Lie algebras
Let be a simple algebraic group defined over an algebraically closed
field of characteristic and let \g be the Lie algebra of . It is
well known that for large enough the spectrum of the cohomology ring for
the -th Frobenius kernel of is homeomorphic to the commuting variety of
-tuples of elements in the nilpotent cone of \g
[Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{10} (1997), 693--728].
In this paper, we study both geometric and algebraic properties including
irreducibility, singularity, normality and Cohen-Macaulayness of the commuting
varieties C_r(\mathfrak{gl}_2), C_r(\fraksl_2) and where is
the nilpotent cone of \fraksl_2. Our calculations lead us to state a
conjecture on Cohen-Macaulayness for commuting varieties of -tuples.
Furthermore, in the case when \g=\fraksl_2, we obtain interesting results
about commuting varieties when adding more restrictions into each tuple. In the
case of \fraksl_3, we are able to verify the aforementioned properties for
C_r(\fraku). Finally, applying our calculations on the commuting variety
C_r(\overline{\calO_{\sub}}) where \overline{\calO_{\sub}} is the closure
of the subregular orbit in \fraksl_3, we prove that the nilpotent commuting
variety has singularities of codimension .Comment: To appear in Journal of Pure and Applied Algebr
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