Classical quotient rings of generalized matrix rings (Q1805128)

Let \(R\) be a generalized matrix ring, i.e. \(R\) is a ring with a set \(E\) of orthogonal idempotents which add to 1. Suppose further that whenever \(e\) and \(f\) are elements of \(E\) and \(x\) is a non-zero element of \(eRf\), then \(xy\) is non-zero for some element \(y\) of \(fRe\). It was shown by Wauters and Jespers that \(R\) is semi-prime if and only if \(eRe\) is semi- prime for all \(e\) in \(E\). It is shown here that \(R\) is semi-prime right Goldie if and only if \(eRe\) is semi-prime right Goldie for all \(e\) in \(E\), and when this happens the quotient ring of \(R\) is determined in terms of the quotient rings of the rings \(eRe\) and the modules \(eRf\) for \(e\) and \(f\) in \(E\). Minor confusion may be caused to the reader by the fact that one of the pages has been printed twice.