Let $R$ be a commutative ring with nonzero identity. An $R$-module $M$ is called $\phi$-P-flat if $x \in \Ann(s)M$ for every non-nilpotent element $s R$ and $x\in M$ such that $sx=0$. In this paper, we introduce study the class of $\phi$-PF-rings, i.e., rings in which all ideals are $\phi$-P-flat. Among other results, transfer $\phi$-PF-ring to amalgamation investigated. Several examples deline...

background and objectives: pomegranate juice (pj) contains large particles that stick to evaporator walls causing off flavors in the concentrate due to burning. microfiltration is used to clarify pj. fouling is a limiting phenomenon that can prevent the industrialization of membrane clarification. changes in the geometry of the membrane module such as using baffles are useful to decrease this p...

Let R be a right GF -closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the G...

We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if Torn (Ω, k) = 0 for all n = 1, . . . , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor1 (Ω, k) = ...

Background and Objectives: Pomegranate juice (PJ) contains large particles that stick to evaporator walls causing off flavors in the concentrate due to burning. Microfiltration is used to clarify PJ. Fouling is a limiting phenomenon that can prevent the industrialization of membrane clarification. Changes in the geometry of the membrane module such as using baffles are useful to decrease this p...

Let G be a finite group acting linearly on a vector space V over a field K of positive characteristic p, and let P ≤ G be a Sylow p-subgroup. Ellingsrud and Skjelbred [20] proved the lower bound depth(K[V ]) ≥ min{dim(V P ) + 2,dim(V )} for the depth of the invariant ring, with equality if G is a cyclic p-group. Let us call the pair (G,V ) flat if equality holds in the above. In this paper we u...

The lamination of OPV modules to corrugated roof cladding has been undertaken. The 3-dimensional form of the cladding provides three advantages for outdoor OPV deployment; firstly the ‘footprint’ of the solar cell is reduced, which leads to B10% improved power conversion (PCE) efficiency per unit area. Secondly, the oblique angle performance is enhanced, leading to increased output in the early...

Given a Laurent polynomial over flat $$\mathbf {Z}$$ -algebra, Vlasenko defines formal group law. We identify this law with coordinate system of functor. When the “Hasse–Witt matrix” is invertible, matrix by taking certain $$p$$ -adic limit. show that Frobenius Dieudonné module modulo .

Let R be a ring. A right R-module is said to C-flat if the kernel of any epimorphism B → C-pure in B, i.e. induced map Hom(C,B) Hom(C,A) surjective for cyclic C. Projective modules are and weakly-flat neat-flat. In this article, it discussed connections between C-flat, weakly-flat, neat-flat singly flat modules. It shown that coincide with singly-projective over arbitrary rings. Next, several c...