On normal subgroups of division rings which are radical over a proper division subring. (Q2875473)

Let \(F\) be a field and \(D\) an associative division ring with centre \(Z(D)=F\). We say that \(D\) is centrally finite, if the dimension \([D:Z(D)]\) is finite; \(D\) is called locally finite, if every finite subset of \(D\) generates a finite-dimensional \(F\)-subalgebra of \(D\). By a Kurosh element of \(D\), we mean a noncentral element \(x\in D\) that is contained in \(D_x\setminus Z(D_x)\), for some division subring \(D_x\) of \(D\) with \([D_x:Z(D_x)]<\infty\). Clearly, the complement \(D\setminus F\) consists of Kurosh elements whenever \(D\) is locally finite over \(F\); the same holds in case \(D\) is algebraic and \(F\) is a local field [see \textit{I. D. Chipchakov}, J. Algebra 160, No. 2, 342-379 (1993; Zbl 0818.16019)]. Moreover, \(D_x\) can be chosen so that \(Z(D_x)=F\), for every \(x\in D\setminus F\), provided that \(F\) satisfies the following condition for each prime number \(p\): there exists a positive integer \(m_p\) with \(\text{Brd}_p(F')\leq m_p\), for every finite extension \(F '/F\), where \(\text{Brd}_p(F ')\) is the Brauer \(p\)-dimension of \(F '\), in the sense of \textit{A. Auel, E. Brussel, S. Garibaldi}, and \textit{U. Vishne} (see [Transform. Groups 16, No. 1, 219-264 (2011; Zbl 1230.16016)], and for a proof, the reviewer's paper [in Mosc. Univ. Math. Bull. 43, No. 2, 18-21 (1988); translation from Vestn. Mosk. Univ., Ser. I 1988, No. 2, 15-17 (1988; Zbl 0657.16010)]). It is worth noting that this condition is satisfied by global fields and local fields (by class field theory), and by function fields of algebraic varieties defined over a finite or an algebraically closed field \(F_0\) (see \textit{Matzri}'s preprint [``Symbol length in the Brauer group of a field'', available online at \url{arxiv:1402.0332v1}]).NEWLINENEWLINE The paper under review shows that there exists a non-algebraic division ring \(\Delta\), such that all \(x\in\Delta\setminus Z(\Delta)\) are Kurosh elements. Its main result states that if \(K\) is a division subring of the division ring \(D\), which includes \(F=Z(D)\), and if \(N\) is a normal subgroup of \(D^*\) that is radical over \(K\), i.e. for each \(h\in N\), we have \(h^{m(h)}\in K\), for some positive integer \(m(h)\), then the set \(N\setminus K\) does not contain any Kurosh element. Using this result and the Herstein-Scott theorem [see \textit{I. N. Herstein, W. R. Scott}, Can. J. Math. 15, 80-83 (1963; Zbl 0109.02602)], the authors prove that \(N\subseteq F^*\), under the extra hypothesis that \(D\) is locally finite and \(K\neq D\). When \(D\) is arbitrary and \(K\neq D\), it is not known that the inclusion \(N\subseteq F '\) retains validity; the question is open even in case \(K=F\) (Herstein's Conjecture). It is proved that the assertion is no longer true, if \(K\) is merely a proper subring of \(D\); in this case, a normal subgroup of \(D^*\) radical over \(K^*\) need not be commutative.