Abstract
With the aid of ultrafilters an example is constructed of a primitive ring with involution and zero socle in which the symmetric elementsS satisfys2c=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.
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