Abstract
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.
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