Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.
In this paper, we study some algebras [Formula: see text] whose unit groups [Formula: see text] or subnormal subgroups of [Formula: see text] are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated.
AbstractLet $D$ be a division ring whose group of units satisfies a non-trivial group identity $w$. Let $\alpha$ be the sum of positive degrees of indeterminates occurring in $w$. If the centre of $D$ contains more than $3\alpha$ elements, then $D$ is commutative.AMS 2000 Mathematics subject classification: Primary 16R50. Secondary 16K20
On division subrings normalized by almost subnormal subgroups in division rings
