Algebraic commutators with respect to subnormal subgroups in division rings

Abstract

Let \(D\) be a division ring and \(K\) a subfield of \(D\) which is not necessarily contained in the center \(F\) of \(D\). We study the structure of \(D\) under the condition of left algebraicity of certain subsets of \(D\) over \(K\). Among others, it is proved that if \(D^*\) contains a noncentral normal subgroup which is left algebraic over \(K\) of bounded degree \(d\), then \([D:F]\le d^2\). In case \(K=F\), the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of \(D^*\) are algebraic of bounded degree \(d\) over \(f\), then \([D:F]\le d^2\).