Annihilators in rings with involution

In this paper we examine the annihilators of algebraic expressions in the symmetric, or skew-symmetric, elements of a ringR with involution. LetL be the set of elements ofR each of which is annihilated on the right by some power of each symmetric element. IfR contains no nonzero nil left ideal thenL=0. The subset ofL annihilated by a fixed power of each symmetric element is always a nil left ideal ofR of bounded index. WhenR is an algebra over a fieldF, letA be the left ideal of elements annihilated on the right by some polynomial in each symmetric element. If eitherF is uncountable or the degrees of the polynomials are bounded, thenA generates an algebraic ideal ofR. Similar results are obtained by using skew-symmetric elements instead of symmetric elements.