The primeness of noncommutative polynomials on prime rings.

We study the primeness of noncommutative polynomials on prime rings. Let R be a prime ring with extended centroid C, ρ a right ideal of R, f(X1,…,Xt) a noncommutative polynomial over C, which is not a polynomial identity (PI) for ρ, and a,b ∈ R∖{0}. Then af(x1,…,xt)b = 0 for all x1,…,xt ∈ ρ if and only if one of the following holds: (i) aρ = 0; (ii) ρC = eRC for some idempotent e ∈ RC and b ∈ ρC such that either f(X1,…,Xt)Xt+1 is a PI for ρ or f(X1,…,Xt) is central-valued on eRCe and ab = 0. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of X-primeness of rings. [ABSTRACT FROM AUTHOR]