On the subring generated by the symmetric elements

Abstract

J. M. Osborn [3] has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S-, the subring generated by S (the set of symmetric elements), is A, the ring itself. In this paper we raise the question as to when S=A under a weaker hypothesis and prove the following theorem. (A henceforth is such that 2A =A and 2x=@ implies x=@ for any xCA.)