Decompositions of Quotient Rings and m -Power Commuting Maps.

LetRbe a semiprime ring with symmetric Martindale quotient ringQ,n ≥ 2 and letf(X) = Xnh(X), whereh(X) is a polynomial over the ring of integers withh(0) = ±1. Then there is a ring decompositionQ = Q1⊕Q2⊕Q3such thatQ1is a ring satisfyingS2n−2, the standard identity of degree 2n − 2,Q2 ≅ Mn(E) for some commutative regular self-injective ringEsuch that, for some fixedq > 1,xq = xfor allx ∈ E, andQ3is a both faithfulS2n−2-free and faithfulf-free ring. Applying the theorem, we characterizem-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring. [ABSTRACT FROM AUTHOR]