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 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

The research shed light on the historic evolution of Baghdad through its long, expansive history. The starting point focuses on the geographic characteristics, and the nature of its habitation, prior to laying the circular plan of Baghdad. Then the research proceeds to cover the stage of building the round city of Baghdad. The research continue to cover the expansion and sequential growth across the banks of Tigris river.
A concentrated attention is devoted to analyses the morphological, geographical and above all the makeup of present day city of Baghdad, pinpointing the apathetic plans, decisions, and actions which completely disfigured the image, and tradition of the old city of Baghdad, behind the delusive slogans of “comprehens

n this paper, we formulate three mathematical models using spline functions, such as linear, quadratic and cubic functions to approximate the mathematical model for incoming water to some dams. We will implement this model on dams of both rivers; dams on the Tigris are Mosul and Amara while dams on the Euphrates are Hadetha and Al-Hindya.

A calamitic symmetric liquid crystalline consisting of an azo group containing 5H-Thiazolo[3,4-b][1,3,4]thiadiazole moiety compound[III] was synthesized via sequence reactions starting from reaction terephthaldehyde with mercaptoacetic acid and thiosemicarbazide in the presence of concentrated sulfuric acid to synthesized 5,5'-(1,4-phenylene)bis(5Hthiazolo[4,3-b][1,3,4]thiadiazol-2-amine)[I] then the azo compound [II] synthesized by coupling between diazonium salt of the compound [I] with phenol at (0-4) ̊C., after that the compound [III] was synthesized by the reaction of the compound [II] with methyl bromide in alkaline media. The compounds are characterized by melting points, FTIR and 1HNMR spectroscopy. The mesomorphic behavior was stu

The university Service employee is considered one of the categories of public employees, However, The academic and administrative tasks duties that undertakes have made him a special importance and a distinguished legal status, Therefore, we find That the Iraqi legislatures assigned this category a special Law, it is the university Service law No (23) of 2008 as amended who dealt with some aspects related to serving this category, such as duties, right, condition for appointment and award of academic titles, and financial allocation, such as university service allocation, academic title allocations, and some other Provisions leaving other matters to the general service laws and employee discipline, among the matters that the legislator negl

     Recently, the Internet of Things has emerged as an encouraging technology that is scaling up new heights towards the modernization of real word physical objects into smarter devices in several domains. Internet of Things (IoT) based solutions in agriculture drives farming into a smart way through the proliferation of smart devices to enhanced production with minimal human involvement. This paper presents a comprehensive study of the role of IoT in prominent applications of farming, wireless communication protocols, and the role of sensors in precision farming. In this research article, the existing frameworks in IoT-based agriculture systems with relevant technologies are presented. Furthermore, the comparative analysis of the a