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    On one-sided Lie nilpotent ideals of associative rings

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    We prove that a Lie nilpotent one-sided ideal of an associative ring RR is contained in a Lie solvable two-sided ideal of RR. 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 R.R. One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form [...[[r1,r2],...],rn1],rn][... [ [r_1, r_{2}], ... ], r_{n-1}], r_{n}] are also studied.Comment: 5 page

    Commuting varieties of rr-tuples over Lie algebras

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    Let GG be a simple algebraic group defined over an algebraically closed field kk of characteristic pp and let \g be the Lie algebra of GG. It is well known that for pp large enough the spectrum of the cohomology ring for the rr-th Frobenius kernel of GG is homeomorphic to the commuting variety of rr-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 Cr(N)C_r(\N) where N\N is the nilpotent cone of \fraksl_2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of rr-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 Cr(N)C_r(\N) has singularities of codimension 2\ge 2.Comment: To appear in Journal of Pure and Applied Algebr
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