On prime and primitive ideals (Q2705846)

Let \(R\) be a ring and \(Q\) be a prime ideal. Every of the following conditions is proved to be sufficient for \(Q\) to be primitive: -- \(R\) is semi-simple, its structure space is discrete, \(Q\) is proper and its annihilator is nonzero; -- the structure space of the ring \(R\) is Hausdorff and \(R/Q\) is semi-simple; -- the structure space of the ring \(R\) is granular (i.e. it contains a dense subset consisting of isolated points) and the annihilator of \(Q\) is nonzero; -- \(R\) is semi-prime, the annihilator of \(Q\) is nonzero and contains the annihilator of the socle of \(R\). Except that, several sufficient conditions for the granularity of the structure space of a ring are obtained in the article.