When every $P$-flat ideal is flat

Document Type: Research Paper

Authors

1 Department of Mathematics, FST, University of Fez

2 Department of Mathematics, Faculty of Sciences and Technology, University of Fez, Fez, Morocco.

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=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.

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