Let \(a\) be an element of a ring \(R\). The ring \(R_a\) that is obtained by defining on the Abelian group \((aRa,+)\) the multiplication \(axa\cdot aya=axaya\) is called the local ring of \(R\) at \(a\). This concept was introduced by K. Meyberg in 1972 in the nonassociative context of Jordan systems.
The primary purpose of the paper under review is to characterize those elements \(a\in R\) for which forming the local ring at \(a\) and forming a quotient ring \(Q\) are interchangeable operations, that is, to determine under what conditions \(Q(R)_a\cong Q(R_a)\), where \(Q(R)\) stands for the maximal ring of left quotients of \(R\), the Martindale symmetric ring of quotients, or the maximal symmetric ring of quotients.
It turns out that in case \(R\) is semiprime the two operations commute for all three types of quotient rings precisely when \(a\) is von Neumann regular in \(Q(R)\), that is, if there exists \(q\in Q(R)\) such that \(aqa=a\). As a corollary, it is shown that if \(R\) is a prime ring and \(R_a\) is a PI-ring for some \(a\in R\), then the maximal ring of left quotients \(Q^\ell_{\max}(R)\) of \(R\) is a primitive ring with nonzero socle.