m-Power Commuting MAPS on Semiprime Rings

Abstract

Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 − e)U ≅ M₂(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting.

Key Words:

• (Commuting) Centralizing map
• Derivation
• Faithful g-free ring
• Maximal right ring of quotients
• m-Power commuting map
• Orthogonally complete
• Semiprime ring

2010 Mathematics Subject Classification:

• 16N60
• 16W25
• 16R60
• 16D50

ACKNOWLEDGMENTS

The authors are grateful to the referee for carefully reading their manuscript. The work was carried out when the first author was visiting NCTS/TPE of Taiwan and when the third author was visiting Gebze Institute of Technology sponsored by TUBITAK. The first author gratefully acknowledges the support from NCTS/TPE of Taiwan and kind hospitality from the host university, National Taiwan University. The third author gratefully acknowledges the support from TUBITAK and kind hospitality from the host university. The research of the third author was supported by NSC and NCTS/TPE of Taiwan.

Notes

Communicated by M. Bre[sbreve]ar.

The authors are Members of Mathematics Division, NCTS (Taipei Office).