In this paper‎, ‎we study the Kropina transformation of exponential (α,β)-metric F=α\exp(s),  s:=β/α‎. ‎We characterize the conditions under which this class of (α,β)-metric is locally projectively flat‎, ‎locally dually flat‎, ‎and Douglas metric‎. ‎Based on‎, ‎we show that the Kropina transformation of an exponential (α,β)-metric is locally projectively flat‎, ‎locally dually flat and Douglas metric if and only if the exponential (α,β)-metric is locally projectively flat‎, ‎locally dually flat and Douglas metric‎, ‎respectively.

In this paper, we first study a special class of (α,β)-metrics in the form F = α + εβ + k β2/α , where α is Riemannian metric, β is a 1-form, and ε,k(≠ 0) are constant. We give a complete classification for such metrics to be locally dually flat. By assumption β is a conformal 1-form, we show that the metric is locally dually flat if and only if α is a Euclidean metric and β is a constant 1-form. Further, we classify locally dually flat of a class of Finsler metric in the form F = α exp( α/β ) + εβ, where ε is constant.

In this paper, we consider the generalized Kropina conformal change of m-th root metric and for this, prove a necessary and su, cient condition of locally projectively atness. Also we proved a necessary and su, cient condition for the generalized Kropina conformal change of m-th root metric is locally dually at.

This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element RDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras -the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of our main theorem, we present (equational) axiomatizations for several subvarieties of RDQDStSH1. The paper concludes with some open problems for further investigation.

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended V-De Morgan law introduced in [20]. Then, using this result and the results of [20], we prove our main result which gives an explicit description of simple algebras (=subdirectly irreducibles) in the variety of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety.In Part II, we prove, using the description of simples obtained in this Part, that the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element RDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras {the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of RDQDStSH1. The Part II concludes with some open problems for further investigation.

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 Prufer domain. We provide necessary and sufficient conditions for R=A µE 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 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.