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.

Key Words:

• Derivation
• Faithful f-free ring
• Linear differential polynomial
• m-Power commuting map
• Semiprime ring
• Symmetric Martindale quotient ring

2000 Mathematics Subject Classification:

• 16N60
• 16W25
• 16R50

ACKNOWLEDGMENTS

Part of the work was carried out when the third author was visiting Gebze Institute of Technology sponsored by TUBITAK (Turkey). He gratefully acknowledges the support from TUBITAK and kind hospitality from the host university. The third author of the work was supported by NSC and NCTS/TPE of Taiwan.

Notes

Communicated by M. Bresar.

Dedicated to Professor P.-H. Lee on the occasion of his retirement.