WHEN EVERY P-FLAT IDEAL IS FLAT

Abstract. In this paper, we study the class of rings in which every P- at ideal is at and which will be called PFF-rings. In particu-lar, Von Neumann regular rings, hereditary rings, semi-hereditary ring, PID and arithmetical rings are examples of PFF-rings. In the context domain, this notion coincide with Prufer domain. We provide necessary and sufficient conditions for R = A / E to be a PFF-ring where A is a domain and E is a K-vector space, where K: = qf(A) or A is a local ring such that ME: = 0. We give examples of non-fqp PFF-ring, of non-arithmetical PFF-ring, of non-semihereditary PFF-ring, of PFF-ring with wgldim> 1 and of non-PFF Prufer-ring. Also, we investigate the stability of this property under localization and homomorphic image, and its trans-fer to nite direct products. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property