Author/Authors :
Cheniour، Fatima نويسنده University S. M. Ben Abdellah , , Mahdou، Najib نويسنده University S. M. Ben Abdellah ,
Abstract :
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=A\propto 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.
