Some Results On Rings Of Quotients

For a general ring R, there exists the maximal ring of left quotientsQmaxl(R), the maximal symmetric ring of quotientsQσ(R); for a semiprime ring R, there ex-ists Martindale two-sided ring of quotientsQr(R), Martindale symmetric ring of quotientsQs(R). And We have known those elements in a semiprime ring R for which taking local rings at elements and the maximal ring of left quotients Q are commut-ing operations. Of course, if we instead a of the idempotent elemente, it must be true that the maximal left quotients ring of a cornerQmaxl(eRe) and the corner of the maximal left quotient ring eQmaxl(R)e are isomorphic. However G. Aranda Pino, M. A. Gomez Lozano and so on prove that under conditions of regularity, the maximal left quotient ring of a corner of a ring is the corner of the maximal left quotient ring. In this paper,on the maximal symmetric ring of quotients, we obtain similar results adding some conditions. We also prove that those elements in a semiprime ring R for which taking local rings at elements and the Martindale two-sided ring of quotients are commuting operations. That is to say 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 the Martindale two-sided ring of quotients Qr(R) at α. If R and S are two non-unital Morita equivalent rings, then their maximal left quotient rings are not necessarily Morita equivalent. However we prove that the ideals generated by two Morita equivalent idempotent rings inside their own maximal symmetric left quotient ring are Morita equivalent.