Strongly prime ideals in CS-rings (Q2703114)

A right ideal \(I\) of a ring \(R\) with 1 is said to be strongly prime if whenever \(a,b\in R\) with \(ab\in I\) and \(aIb\subseteq I\) then \(a\in I\) or \(b\in I\). In particular, every maximal right ideal is strongly prime. It is shown that a projective strongly prime right ideal of a von Neumann regular right CS ring \(R\) is a direct summand of \(R\), thus extending the known result in which ``CS'' is replaced by ``self-injective''. Also if \(I\) is a projective strongly prime right ideal of a right continuous ring \(R\) and \(I\) contains the Jacobson radical \(J\) of \(R\), then \(I=eR+J\) for some idempotent element \(e\) of \(R\).