Herstein's question about simple rings with involution

TL;DR

This paper investigates Herstein's question on simple rings with involution, establishing conditions under which the ring equals the square of the symmetric elements, and confirms the positive answer for matrix rings.

Contribution

It provides new criteria for when a simple ring with involution satisfies R=S^2, extending known results to broader classes of rings.

Findings

In such rings, R=S^3.

Criteria involving elements with non-zero commutators or anticommutators determine R=S^2.

Results hold without restrictions on the dimension over the center.

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.