The radical trace property and primary ideals (Q1815018)

The following statements are equivalent for a Prüfer domain \(R\): (1) the ideal \(\{fm |f\in \Hom_R(M,R)\), \(m\in M\}\) is radical for any \(R\)-module \(M\) (shortly \(R\) is a \(RTP\)-domain), (2) for each primary ideal \(Q\), either \(Q\) is invertible, or \(QQ^{-1}\) is a prime ideal, (3) for each primary ideal \(Q\), if \(Q^{-1}\) is a ring, then \(Q\) is prime, (4) each branched prime is the radical of a finitely generated ideal. Applications, examples and questions are enclosed.