Let be a commutative ring with unity and let be a non-zero unitary module. In
this work we present a -small projective module concept as a generalization of small
projective. Also we generalize some properties of small epimorphism to δ-small
epimorphism. We also introduce the notation of δ-small hereditary modules and δ-small
projective covers.

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

The objective Effect of Internal and External Environment and its Psychological & Practical Reflection on the Political Decision-Making Process

    Form recurrence of financial crises phenomenon disturbing and attention , and returns the reasons so that its negative effects were sharp and dangerous , because of the nature and cause of Ncaha , threatened political and economic stability of the countries in which they occur these crises , in addition to Machmlh these crises spread of contagion across multiple channels to include other countries many developed and developing , and the reason for this to the openness of the economic and financial witnessed by the countries affected by crises and other countries concerned, the financial crisis is a case of financial turmoil appears in one of the sections of the financial system one and extends to

Absitract

This research was interested in studying the phrase “I don’t know” in the Lisan al-Arab dictionary, and Lisan al-Arab was based on collecting its material from five dictionaries, including: Tahdhib al-Lugha, al-Muqamah, al-Sahih, and the footnotes of Ibn Berri, al-Nihaya and Gharib al-Hadith. The objection to this phrase, and the discussion of its various implications among linguists and the clarification of the closest and most famous content to it according to the data presented to the researcher in his research journey, and to reach this goal, the research division into a preface, five demands and a conclusion and followed the list of sources and references. To define the lexicon of Lisan al-Ar

The fluctuations in oil prices in world markets affect the general budget and the trade balance of the rent countries, because oil is a strategic commodity affected by economic and political factors. The fluctuations in oil prices affect the public budgets of the rent countries through the public revenue side of oil revenues. On the other hand, these fluctuations affect the balance of trade through the volume of oil exports, which lead to imbalance of trade surplus or deficit .                                           &nbs