Annihilators in rings with involution (Q1061199)

Let R be a ring with involution \({}^*\), \(S=\{r\in R|\) \(r^*=r\}\) and \(K=\{r\in R|\) \(r^*=-r\}\). Let L be the set of elements of R each of which is annihilated on the right by some power of each element of S. It is shown that if R contains no nonzero nil left ideal, then \(L=0\). Further the subset of L annihilated by a fixed power of each element of S is always a nil left ideal of bounded index. When R is an algebra over a field F, let A be the left ideal consisting of elements annihilated on the right by some polynomial in each element of S. If either F is uncountable or the elements of S are algebraic of bounded degree, then A generates an algebraic ideal of R. Similar results are obtained by using the elements of K, assuming that R has no nonzero commutative ideal.