The primeness of noncommutative polynomials on prime rings

We study the primeness of noncommutative polynomials on prime rings. Let Formula: see text be a prime ring with extended centroid Formula: see text, Formula: see text a right ideal of Formula: see text, Formula: see text a noncommutative polynomial over Formula: see text, which is not a polynomial identity (PI) for Formula: see text, and Formula: see text. Then Formula: see text for all Formula: see text if and only if one of the following holds: (i) Formula: see text; (ii) Formula: see text for some idempotent Formula: see text and Formula: see text such that either Formula: see text is a PI for Formula: see text or Formula: see text is central-valued on Formula: see text and Formula: see text. 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 Formula: see text-primeness of rings.