When every $P$-flat ideal is flat

In this paper, we study the class of rings in which every $P$-flat ideal is flat and which will be called $PFF$-rings. In particular, 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 Pr«{u} fer domain. We provide necessary and sufficient conditions for $R=Apropto 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-$PFF$ Pr» {u} fer-ring. Also, we investigate the stability of this property under localization and homomorphic image, and its transfer to finite direct products. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.