Herstein's question about simple rings with involution

Abstract

The aim of this paper is to try to answer Herstein's question concerning simple rings with involution, namely: If $R$ is a simple ring with an involution of the first kind, with $dim_{Z(R)}R > 4$ and $\Char(Z(R))\neq 2$, is it true that $S^2=R$? We shall see that in such a ring $R$, $R=S^3$. We shall bring two possible criteria, each shows when $R=S^2$. The first criterion: There exist $x,y \in S$ such that $xy-yx \neq 0$ and $xSy \subseteq S^2$ $\Leftrightarrow$ $S^2=R$. The second criterion: There exist $x,y \in S$ such that $xy+yx \neq 0$ and $xKy \subseteq S^2$ $\Leftrightarrow$ $S^2=R$. Actually, those results are true without any restriction on the dimension of $R$ over $Z(R)$. In the special case of matrices (with the transpose involution and with the symplectic involution) over a field of characteristic not equal to 2, it is not difficult to find, for example, $x,y \in S$ such that $xy-yx \neq 0$ and for every $s \in S$, $xsy \in S^2$. Therefore, proving Herstein's remark that for matrices the answer is known to be positive. Similar results for $K^6$, $K^4$, $K+KSK$, $KS+K^2$, $SKS$ and $S^2K$ can also be found.