Local Rings of Rings of Quotients

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Abstract

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a.

Keywords

Semiprime ring Local ring at an element Ring of quotients PI-element 

Mathematics Subject Classifications (2000)

16E60 16E50 16R20 
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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Departamento de Álgebra, Geometría y TopologíaUniversidad de MálagaMálagaSpain

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