Lie nilpotence of group rings

Group rings whose symmetric elements are Lie nilpotent

“…By Lemma 5, some positive power of x is central, and therefore |ζ 2 | = ∞. By the corollary in [1], we are done. Thus, we may assume that G is a torsion nilpotent group.…”

“…If Q 8 ⊆ G, then our proof is quite similar to that of [1]. or b −1 a, then Lemma 3 gives us a contradiction.…”

“…Since (2) certainly implies (1), and the equivalence of (2) and (3) is given by [5,Theorem V.4.4], it will suffice if we can show that (1) implies (2).…”

“…In [4], Passi, Passman and Sehgal showed that F G is Lie nilpotent if and only if G is nilpotent and p-abelian, where char F = p. (Recall that G is p-abelian if G is a finite p-group, and 0-abelian will be taken to mean abelian.) In [1], Giambruno and Sehgal showed that if G contains no 2-elements, then (F G) + is Lie nilpotent if and only if F G is Lie nilpotent. We cannot expect this to remain true if G has 2-elements.…”

Antisymmetric Elements in Group Rings Ii

“…Furthermore, if F is a nonabsolute field then U ϕ (A) does not contain a free group of rank 2 if and only if (A) − ϕ is commutative. Giambruno and Sehgal, in [6], showed that if B is a semiprime ring with involution ϕ, B = 2B and (B) − ϕ is Lie nilpotent then (B) − ϕ is commutative and B satisfies a polynomial identity of degree 4. Special attention has been given to the classical involution * on RG, that is, the R-linear map defined by mapping g ∈ G onto g −1 .…”