Decompositions of Quotient Rings and m-Power Commuting Maps

Abstract

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X n h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q 1 ⊕ Q 2 ⊕ Q 3 such that Q 1 is a ring satisfying S 2n−2, the standard identity of degree 2n − 2, Q 2 ≅ 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 Q 3 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.