Lie rings and the Engel condition

Engel-5 Lie Algebras

Group rings satisfying generalized Engel conditions

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group an...

Group Rings with Solvable «-engel Unit Groups'

Let KG be the group ring of a group G over a field of characteristic p > 0, p ^ 2, 3. Suppose G contains no element of order p (if p > 0). Group algebras KG with unit group U(KG) solvable and n-Engel are characterized. Let ATG be the group ring of a group G over a field K of characteristic p > 0 and let U(KG) denote its group of units. Several authors including Bateman [1], Bateman and Coleman ...

Lie Ideals and Generalized Derivations in Semiprime Rings

Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R ! R is called a generalized derivation on R if there exists a derivation d : R to R such that F(xy) = F(x)y + xd(y) holds for all x y in R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.

Generalized Skew Derivations with Engel Conditions on Lie Ideals

Let R be a prime ring and L a noncommutative Lie ideal of R. Suppose that f is a nonzero right generalized β-derivation of R associated with a β-derivation δ such that [f(x), x]k = 0 for all x ∈ L, where k is a fixed positive integer. Then either there exists s ∈ C scuh that f(x) = sx for all x ∈ R or R ⊆ M2(F ) for some field F . Moreover, if the latter case holds, then either charR = 2 or cha...