On the unit groups of rings with involution (Q2151112)

Let \(R\) be a ring with involution \(\star\) of the first kind, i.e., \(x^{\star}=x\) for all \(x\) in the center \(Z(R)\) of \(R\). For a subset \(A\) of \(R\) let \(A_+=\{x\in A\mid x^{\star}=x\}\), \(T_A=\{x+x^{\star}\mid x\in A\}\) and \(N_A=\{xx^{\star}\mid x\in A\}\) be, respectively, the sets of symmetric elements in \(A\), the traces and the norms of the elements of \(A\). The leading idea of the paper under review is to study the influence of some properties of the unit group \(R^{\ast}\) on the algebraic structure of \(R\) provided that \(R\) is an artinian or semiprimitive rings with special attention to the case when \(R\) is a division ring. The first group of results is related with the theorem of Cartan-Brauer-Hua that for a division ring \(D\) and its division subring \(A\) the condition \(xAx^{-1}\subseteq A\) for all \(x\in D^{\ast}\) implies that either \(A\subseteq Z(D)\) or \(A=D\) and its \(\star\)-analogue by Herstein. The authors prove that if \(D\) is a division ring with involution and \(A\) is a division subring of \(D\), the subgroup \(G\) of \(D^{\ast}\) is \(\star\)-invariant and noncentral, \(N_G\) is noncentral and \(A\) is \(N_G\)-invariant, then either \(A\subseteq Z(D)\) or \(A=D\). A similar result is established replacing \(N_G\) by \(G_+\). The authors present examples to compare the differences between the effect of the properties of \(G_+\) and \(D^{\ast}_+\) on the structure of \(D\). When the center \(F\) of \(D\) is infinite, \(\text{\text char}(D)\not=2\), \(G\) is a noncentral normal subgroup of \(D^{\ast}\) and one of the sets \(S=T_G\) or \(S=N_G\) is commutative, then \(S\) is contained in \(F\), \(\dim_FD=4\) and \(\star\) is of the symplectic type. In the general case (for any center \(F\) and any characteristic of \(D\)) the same holds when the set \(T_G\cup N_G\) is commutative. The second group of results deals with artinian and semiprimitive rings. If \(R\) is a simple artinian ring with involution and \(G\) is a noncentral normal subgroup of \(R^{\ast}\), then the condition that \(T_G\cup N_G\) is in the center \(F\) of \(R\) implies that either \(R\) is a division ring of order 2 or \(R\) is isomorphic to the \(2\times 2\) matrix ring \(M_2(F)\) and \(\star\) is of the symplectic type. Finally, let \(R\) be an algebra over a field \(F\) and let \(T_{R^{\ast}}\) and \(N_{R^{\ast}}\) be in \(F\). If \(R\) is semiprimitive and algebraic over \(F\), then \(R\) is a subdirect product of division rings of index 2 and matrix algebras of order \(\leq 2\). If \(R\) is artinian, then it is a finite direct product of division rings of index 2 and matrix algebras of order \(\leq 2\) over fields.