Commutators with Power Central Values on a Lie Ideal
نویسندگان
چکیده
Theorem ([11, Theorem 2, page 282]). Let R be a prime ring, L a noncommutative Lie ideal of R and d 6= 0 a derivation of R. If [d(x), x] ∈ Z(R), for all x ∈ L, then either R is commutative, or char(R) = 2 and R satisfies s4, the standard identity in 4 variables. Here we will examine what happens in case [d(x), x]n ∈ Z(R), for any x ∈ L, a noncommutative Lie ideal of R and n ≥ 1 a fixed integer. One cannot expect the same conclusion of Lanski’s theorem as the following example shows: Example 1. Let R = M2(F ), the 2 × 2 matrices over a field F , and take L = R as a noncommutative Lie ideal of R. Since [x, y]2 ∈ Z(R), for all x, y ∈ R, then also [d(x), x]2 ∈ Z(R), for all x ∈ R.
منابع مشابه
Some properties of nilpotent Lie algebras
In this article, using the definitions of central series and nilpotency in the Lie algebras, we give some results similar to the works of Hulse and Lennox in 1976 and Hekster in 1986. Finally we will prove that every non trivial ideal of a nilpotent Lie algebra nontrivially intersects with the centre of Lie algebra, which is similar to Philip Hall's result in the group theory.
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