Primitive rings with involution and pivotal monomials

With the aid of ultrafilters an example is constructed of a primitive ring with involution and zero socle in which the symmetric elementsS satisfys ² c=s,c depending on and commuting withs (thusS satisfies the pivotal monomialx). However, by suitably restricting the definition of generalized pivotal monomial (essentially by making the variables linear) it is shown that a primitive ring with involution has nonzero socle if and only if the symmetric elements satisfy a restricted GPM.