On dependent elements in rings

out, R will represent an associative ring with center Z(R). As usual the commutator xy−yx will be denoted by [x,y]. We will use basic commutator identities [xy,z] = [x,z]y+x[y,z] and [x,yz]= [x,y]z+y[x,z]. Recall that a ring R is prime if aRb = (0) implies that a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping x x∗ on a ring R is called involution in case (xy)∗ = y∗x∗ and x∗∗ = x hold for all x,y ∈ R. A ring equipped with an involution is called a ring with involution or ∗-ring. An additive mapping D : R → R is called a derivation in case D(xy) = D(x)y+xD(y) holds for all pairs x,y ∈ R. A derivation D is inner in case there exists a ∈ R such that D(x) = [a,x] holds for all x ∈ R. An additive mapping T : R → R is called a left centralizer in case T(xy) = T(x)y is fulfilled for all pairs x,y ∈ R. This concept appears naturally in C∗-algebras. In ring theory it is more com-