Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = Xⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q₁ ⊕ Q₂ ⊕ Q₃ such that Q₁ is a ring satisfying S_2n-2, the standard identity of degree 2n - 2, Q₂ ≅ M_n(E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x^q = x for all x ∈ E, and Q₃ is a both faithful S_2n-2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.