Flat Modules Research Articles

Gorenstein projective, injective, flat modules and dimensions over polynomial rings

We investigate Gorenstein projective, injective, flat modules and dimensions over a polynomial ring using the equivalent category of the module category over the polynomial ring. Several estimates of Gorenstein homological dimensions of modules over the polynomial ring are obtained.

Comprehensive evaluation of integrating radiative sky cooling with compound parabolic concentrator for cooling flux amplifying

The low cooling density of RC technology makes it challenging to meet the cooling requirements of buildings, particularly during the day with intense solar radiation. To address this challenge, a novel RC module coupled with a compound parabolic concentrator (CPC-RC module) was proposed. In this paper, a universal modelling approach is used to characterize the cooling performance of CPC-RC module, showing 18.1% higher than that of flat module during nighttime. However, the CPC's solar concentrating characteristic impacts the RC emitter's daytime performance. To tackle this issue, exploring the optimal tilt angle of CPC-RC module on typical days to eliminate the influence of solar concentrating property is needed. The results indicate that on 21st June, average cooling power of 121.29 W/m2 is achieved when the module is tilted 25° to the anti-sunward side, 15.7% higher than the module placed horizontally. Additionally, this paper shows that the cooling performance of CPC-RC module with non-ideal emitter is still higher than that of flat one, demonstrating CPC's ability to mitigate the rigorous spectral requirements for emitters. This CPC-RC module offers good cooling effect and shows its great potential for integration into buildings, offering a new research direction for the wider application of RC technology.

Performance of spherical and flat solar cells under different environmental conditions

This study investigates the performance degradation of spherical and flat solar modules due to dust accumulation. To isolate the effect of dust, both modules were subjected to identical environmental conditions, including irradiance, temperature, and humidity. The results reveal a significant reduction in power efficiency as dust density increases, with a nearly linear relationship observed. Spherical solar modules exhibited a minor decrease in efficiency (±3.64%) compared with flat solar modules (±5.46%) under increased dust density, consistent with prior research. Dust accumulation affects solar cells by causing dust deposition, light scattering, and temperature rise. Flat solar panels are more susceptible to dust accumulation due to their larger surface area, which attracts more dust particles. In contrast, spherical cells’ smaller surface area minimises dust build-up, making them less prone to efficiency loss. The study also examined the impact of solar intensity and tilt angle on efficiency. Spherical cells demonstrated greater resilience in varying light conditions and tilt angles, maintaining efficiency across a wider range of scenarios than flat panels. These findings emphasise the importance of considering dust accumulation when designing and maintaining solar panels, with spherical modules offering advantages in dusty environments and under changing sunlight conditions.

Characteristic modules and Gorenstein (co-)homological dimension of groups

In this paper, we examine the Gorenstein dimension of modules over the group algebra kG of a group G with coefficients in a commutative ring k. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying k-module and the Gorenstein dimension of G over k. Our method is based on the notion of a characteristic module for G, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded Z-valued functions on such a group G to characterize the Gorenstein flat ZG-modules, in terms of flat modules, and the Gorenstein injective ZG-modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective ZG-modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group G in Kropholler's class, (a) any Gorenstein projective ZG-module is Gorenstein flat and (b) a ZG-module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.

Triviality criteria for unbounded complexes

We study properties of modules that appear as syzygies of acyclic complexes of projective, injective, flat or flat-cotorsion modules and obtain criteria for these complexes to be contractible or totally acyclic. Our results illustrate the importance of strongly fp-injective modules in the study of these properties. We examine implications of the existence of complete resolutions (in a certain weak sense) and the finiteness of the Gorenstein projective dimension of pure-projective modules. We also use the orthogonality in the homotopy category between complexes of flat modules and pure acyclic complexes of cotorsion modules, in order to study the syzygies of acyclic complexes of flat modules. Finally, we present some applications of our results to group rings, regarding complete resolutions and cohomological periodicity.

N-weak injective and n-weak flat modules under m-semidualizing bimodules

Let [Formula: see text] and [Formula: see text] be rings, the [Formula: see text] two non-negative integers and [Formula: see text] an (faithfully) [Formula: see text]-semidualizing bimodule. In this paper, we introduce, via special super finitely presented module [Formula: see text] with respect [Formula: see text], the concepts of [Formula: see text]-[Formula: see text]-weak injective and [Formula: see text]-[Formula: see text]-weak flat modules, and then we obtain some characterizations of two classes of these modules, namely [Formula: see text] and [Formula: see text] which are larger than that of [Formula: see text]-weak injective and [Formula: see text]-weak flat modules, respectively. Also, we investigate Foxby equivalence relative to these modules. Then over any arbitrary ring, we show some nice properties from the classes [Formula: see text] and [Formula: see text].

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Methylene Blue Degradation Using Non-Thermal Plasma

Methylene blue (C16H18ClN3) dye can be decomposed using non-thermal plasma. However, there is a problem in that the maintenance of electrodes and dielectrics is necessary due to the durability and heat generation problems due to the high temperatures. Therefore, in this study, a comparative experiment was performed between the flat DBD plasma module and the diffuser DBD module under the same conditions. For methylene blue decomposition, the characteristic changes in the air flow rate, ozone production rate, energy consumption rate, and decomposition rate were compared. In the experiment, 7 L water was placed in a 15 L reactor, and measurements were performed for approximately 1 h. We performed the same process by setting the initial methylene blue concentration to 143 mg/L. According to the results, the flat DBD module achieved a decomposition rate of 100% in 40 min, an energy yield of 46.7 g/kWh, and an ozone generation amount of 6.5 g/h. The diffuser DBD module achieved a decomposition rate of 90%, an energy production of 24.6 g/kWh, and an ozone generation of 1.97 g/h in 60 min.

On the ϕ-weak global dimensions of polynomial rings and ϕ-Prüfer rings

This paper focuses on the study of [Formula: see text]-weak global dimensions in the context of polynomial rings and [Formula: see text]-Prüfer rings. We explore new properties of these dimensions and extend the Hilbert syzygy theorem to [Formula: see text]-weak global dimensions of rings. We also determine the [Formula: see text]-weak global dimension for certain types of [Formula: see text]-Prüfer rings. Key concepts such as [Formula: see text]-flat modules, [Formula: see text]-injective modules, and [Formula: see text]-torsion modules are discussed, along with their hereditary properties in PN-rings. This paper includes several theorems and lemmas that provide insights into the [Formula: see text]-weak global dimensions and their implications in the field of ring theory.

Strongly J-n-Coherent rings

Let $R$ be a ring and $n$ a fixed positive integer. A right $R$-module $M$ is called strongly $J$-$n$-injective if every $R$-homomorphism from an $n$-generated small submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be strongly $J$-$n$-flat, if for every $n$-generated small submodule $T$ of a free left $R$-module $F$, the canonical map $V\otimes T\rightarrow V\otimes F$ is monic; a ring $R$ is called left strongly $J$-$n$-coherent if every $n$-generated small submodule of a free left $R$-module is finitely presented; a ring $R$ is said to be left $J$-$n$-semihereditary if every $n$-generated small left ideal of $R$ is projective. We study strongly $J$-$n$-injective modules, strongly $J$-$n$-flat modules and left strongly $J$-$n$-coherent rings. Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.

Treatment of textile wastewater in a single‐step moving bed‐membrane bioreactor: Comparison with conventional membrane bioreactor in terms of performance and membrane fouling

AbstractMembrane bioreactor (MBR) is a promising technology for the treatment of municipal and industrial wastewater, including highly contaminated textile wastewater. However, membrane fouling remains a critical challenge due to reduced flux. This study investigates the efficacy of a moving bed MBR (MB‐MBR) technology for textile wastewater treatment, focusing on chemical oxygen demand (COD) removal, and its impact on mitigating membrane fouling. In a 50‐day study, a conventional MBR (R1) was compared with an MB‐MBR (R2) augmented with free‐floating biocarriers (accounting for 20% of the reactor volume). Both systems used flat sheet ceramic membrane modules. The results indicate that the MB‐MBR achieved superior performance, with COD and colour removal of 89% and 81%, respectively, compared with 87% and 73% in the conventional MBR. Importantly, the introduction of biocarriers eliminated the need for offline physical membrane cleaning in the MB‐MBR. The free‐floating biocarriers lowered transmembrane pressure, reduced capillary suction time and reduced fouling through their scouring action.

Cross-flow photocatalytic membrane reactor (PMR) based on TiO2–AgBr–Ag–NH2-MIL-88B (Fe) /poly (ethersulfone) for enhanced removal of cefixime, anti-fouling performance and permeability

The organic-metallic framework NH2-MIL-88B (Fe) was initially synthesized using a hydrothermal process., then the TiO2–AgBr–Ag–NH2-MIL-88B (Fe) photocatalytic nanoparticles was successfully ready and used as a filler for mixed matrix membrane (MMM) poly Ether sulfone (PES) was added. A photocatalytic membrane reactor (PMR) for the degradation of cefixime aqueous solution (CFX) was fabricated using a PES-TiO2-AgBr-Ag–NH2–MIL-88B (Fe) flat sheet membrane module. FTIR, XRD, FESEM, and DRS analyses were performed to look into the produced nanoparticles' structural characteristics. Hydrophilicity and membrane surface roughness were investigated using the surface contact angle and AFM pictures test. Additionally, Cross-FESEM analysis was used to examine modifications in membrane morphology. By increasing the TiO2–AgBr–Ag–NH2-MIL-88B (Fe) nanoparticles in the polymer matrix, significant improvements were made to the membrane's hydrophilicity, porosity, pure water flux, and CFX degradation. Furthermore, the nanocomposite membranes' antifouling characteristic was assessed while the CFX solution was being filtered. The obtained results show that the flux and removal of CFX solution is improved by adding TiO2–AgBr–Ag–NH2-MIL-88B (Fe) nanoparticles to PES polymer solution up to 1 % by weight. In addition, the obtained results showed that the addition of TiO2–AgBr–Ag–NH2-MIL-88B (Fe) photocatalytic nanoparticles in the PES membrane significantly improves the antifouling properties.

Progress in module design for membrane distillation

There have been tremendous advances in membrane distillation (MD) since the concept was introduced in 1961: new membrane designs and process configurations have emerged, and its commercial viability has been evaluated in several pilot-scale studies. However, its high energy consumption has hindered its commercialization. One of the most promising ways to overcome this obstacle is to develop more energy-efficient membrane modules. The MD research community has therefore developed diverse new module configurations for hollow fiber and flat sheet membranes that increase the thermal energy efficiency of MD by minimizing thermal polarization, increasing mass transfer across the membrane, and improving heat recovery from the condensed vapor. This review summarizes the progress made in the design of hollow fiber and flat sheet membrane modules for MD applications. It begins with a brief introduction to MD and its configurations before describing developments in module fabrication and highlighting key areas where further research is needed.

Relative Gorenstein flat modules and Foxby classes and their model structures

We introduce the concepts of relative (strongly) cotorsion and relative Gorenstein cotorsion modules for a non-necessarily semidualizing module and prove that there exists a unique hereditary abelian model structure where the cofibrations are the monomorphisms with relative Gorenstein flat cokernel and the fibrations are the epimorphisms with relative cotorsion kernel belonging to the Bass class. In the particular case of a semidualizing module, we investigate the existence of abelian model structures on the category of left (right) R-modules where the cofibrations are the epimorphisms (monomorphisms) with kernel (cokernel) belonging to the Bass (Auslander) class. We also show that the class of relative Gorenstein flat modules and the Bass class are part of weak AB-contexts.

Local, colocal, and antilocal properties of modules and complexes over commutative rings

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles.

Dynamically Mapping the Topography and Stiffness of the Leading Edge of Migrating Cells Using AFM in Fast-QI Mode.

Cell migration profoundly influences cellular function, often resulting in adverse effects in various pathologies including cancer metastasis. Directly assessing and quantifying the nanoscale dynamics of living cell structure and mechanics has remained a challenge. At the forefront of cell movement, the flat actin modules─the lamellipodium and the lamellum─interact to propel cell migration. The lamellipodium extends from the lamellum and undergoes rapid changes within seconds, making measurement of its stiffness a persistent hurdle. In this study, we introduce the fast-quantitative imaging (fast-QI) mode, demonstrating its capability to simultaneously map both the lamellipodium and the lamellum with enhanced spatiotemporal resolution compared with the classic quantitative imaging (QI) mode. Specifically, our findings reveal nanoscale stiffness gradients in the lamellipodium at the leading edge, where it appears to be slightly thinner and significantly softer than the lamellum. Additionally, we illustrate the fast-QI mode's accuracy in generating maps of height and effective stiffness through a streamlined and efficient processing of force-distance curves. These results underscore the potential of the fast-QI mode for investigating the role of motile cell structures in mechanosensing.

Torsion pairs and related modules over trivial ring extensions

Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. We first study how to construct torsion pairs over $R\ltimes M$ from torsion pairs over $R$. Some characterizations of finitely generated (presented) modules, flat modules and coherent rings relative to a torsion pair over $R\ltimes M$ are obtained. Then we discuss the transfers of torsion pairs over $R\ltimes M$ to $R$. Finally, some applications are given in Morita context rings.

Sunflower solar tree vs. flat PV module: A comprehensive analysis of performance, efficiency, and land savings in urban solar integration

Solar energy, a prominent form of renewable energy, can be efficiently harnessed through photovoltaic (PV) technology, which has witnessed significant advancements over the past two decades. Solar tree technology has emerged as a solution to several technical challenges associated with PV systems, including land footprint concerns, aesthetic integration, and efficiency optimization. This experimental study compares a traditional flat PV module and a solar tree modeled after a sunflower. The comparative framework encompasses assessments across three distinct tilt angles to evaluate the impact of inclination on energy output. Empirical findings reveal that the sunflower-shaped module can generate 16–23 % more energy than its flat counterpart. Furthermore, temperature assessments indicate that the flat PV module reached a peak temperature of 51 °C, in contrast to the sunflower's maximum of 41 °C. An examination of the ground footprint revealed that the sunflower PV module afforded 85 % land savings relative to the flat PV module. Additionally, the research noted an average temperature reduction of 3 °C in areas shaded by solar trees, offering potential benefits for agricultural applications.

Harnessing the characteristic of compound parabolic concentrators for directional concentration of emitted thermal radiation and solar shielding in building-integrated radiative cooling

Radiative sky cooling (RC) is a promising solution for meeting the growing cooling demands by passively dissipating waste heat into frigid outer space. However, current RC systems suffer from low cooling power density and limited installation flexibility, impeding their effective application as building cooling strategies. To overcome these challenges, a novel concentrated RC system coupled with a compound parabolic concentrator (CPC) is proposed and experimentally studied. The objective is to investigate the effectiveness of the CPC in enhancing the cooling capability of the RC system and the feasibility of achieving all-day RC in unfavorable working conditions when integrated with building roofs. During nighttime experiments in the humid Nottingham region, the CPC-RC system exhibited an average emitter temperature that was 5.83 °C lower than the ambient temperature, representing a 30 % and 13.6 % improvement in RC performance compared to the flat-RC system and trapezoidal-concentrated RC system, respectively. The Photopia optical software simulation indicates that when the modules are tilted to the north, the CPC functions as a solar shield, effectively limiting solar radiation reaching the emitter surface, which is advantageous for conducting daytime RC experiments. In the daytime experiment, the emitter temperature of the CPC-RC module in anti-sunward group was still 1.59 °C lower than the ambient temperature and 5.49 °C lower than that of flat-RC module. At night, the CPC-RC module of the three placement groups all showed the highest RC effect in the same group. The average emitter temperature of the CPC-RC module in the horizontal placement group is 0.6 °C lower than that of the flat RC module. This novel CPC-RC scheme presents a new energy-saving strategy for buildings and showcases its potential for achieving 24-hour RC when integrated into anti-sunward roofs.