Weosay thatotheosubmodules A, B ofoan R-module Moare µ-equivalent , AµB ifoand onlyoif <<_µand <<_µ. Weoshow thatoµ relationois anoequivalent relationoand hasegood behaviorywith respectyto additionmof submodules, homorphismsr, andydirectusums, weaapplyothese resultsotoointroduced theoclassoof H-µ-supplementedomodules. Weosay thatoa module Mmis H-µ-supplementedomodule ifofor everyosubmodule A of M, thereois a directosummand D ofoM suchothat AµD. Variousoproperties ofothese modulesoarepgiven.

    Gangyong Lee, S. Tariq Rizvi, and Cosmin S. Roman studied Dual Rickart modules. The main purpose of this paper is to define strong dual Rickart module. Let M and N be R- modules , M is called N- strong dual Rickart module (or relatively sd-Rickart to N)which is  denoted by M it is N-sd- Rickart if for every submodule A of M and every homomorphism fHom (M , N) , f (A) is a direct summand of N. We prove that for an R- module M , if R is M-sd- Rickart , then every cyclic submodule of M is a direct summand . In particular, if M<

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.

Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and int

In this paper, certain types of regularity of topological spaces have been highlighted, which fall within the study of generalizations of separation axioms. One of the important axioms of separation is what is called regularity, and the spaces that have this property are not few, and the most important of these spaces are Euclidean spaces. Therefore, limiting this important concept to topology is within a narrow framework, which necessitates the use of generalized open sets to obtain more good characteristics and preserve the properties achieved in general topology. Perhaps the reader will realize through the research that our generalization preserved most of the characteristics, the most important of which is the hereditary property. Two t

     Diabetic kidney disease is an illness of the glomerulus that interferes with the glomerular filtration barrier (GFB), which is worked to enable kidney to selective purification of water and solutes in addition to limiting the movement of large macromolecules such as albumin. In the glomerular endothelium, mesangial cells, foot cells, and the brush border of the proximal tubules, ACE-2 is expressed and that the kidneys represent the highest-expressing region of this enzyme. Thus, the current study aimed to evaluate ACE-2 level in this case compared to healthy condition. The study Conducted with 120 male and female ranging in age (30-65) years old. Ninety patients with type 2 diabetes subdivided into three groups on the basis of A

         Recently, emulgel has emerged as one of the most interesting topical preparations in the field of pharmaceutics. In this research clotrimazole was formulated as topically applied emulgel ; different formulas were prepared. The prepared emulgels were evaluated for their physical appearance , rheological behaviour , and in vitro drug release . The influence of the type of gelling agent (carbopol 934 and methyl cellulose), the concentration of both the emulsifying agent (2% and 4% w/w of mixture of span 20 and tween 20) and the oil phase (5% and 7.5% w/w of liquid paraffin) and the type of oil phase (liquid paraffin and cetyl alcohol), on the drug release from the prepared emulgels was invest

The current study focuses on the bacterium Acinetobacter baumannii due to its importance as a nosocomial infections source in addition to its increased resistance against antibiotics. Different clinical and hospital environment samples were collected, and cultured on A. baumannii selective media: Leed Acinetobacter agar and Herellea agar. A. baumannii have been identified by traditional methods, followed by confirmation using molecular identification to detect bla_{oxa-51 like} gene which is considered a diagnostic gene since it is present in genome of all A. baumannii strains. The result was, nineteen bacterial isolates of A.baumannii were obtained, from twenty-seven suspected isolate

This work focuses on the preparation of pure nanocrystalline SnO2 and SnO2:Cu thin films on cleaned glass substrates utilizing a sol-gel spin coating and chemical bath deposition (CBD) procedures. The primary aim of this study is to investigate the possible use of these thin films in the context of gas sensor applications. The films underwent annealing in an air environment at a temperature of 500 ^◦C for duration of 60 minutes. The thickness of the film that was deposited may be estimated to be around 300 nm. The investigation included an examination of the structural, optical, electrical, and sensing characteristics, which were explored across various preparation circumstances, specifically focusing on varied