Let R be a commutative ring with unity and M be a non zero unitary left R-module. M is called a hollow module if every proper submodule N of M is small (N ≪ M), i.e. N + W ≠M for every proper submodule W in M. A δ-hollow module is a generalization of hollow module, where an R-module M is called δ-hollow module if every proper submodule N of M is δ-small (N δ  M), i.e. N + W ≠M for every proper submodule W in M with M W is singular. In this work we study this class of modules and give several fundamental properties related with this concept
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Let M be a weak Nobusawa -ring and γ be a non-zero element of Γ. In this paper, we introduce concept of k-reverse derivation, Jordan k-reverse derivation, generalized k-reverse derivation, and Jordan generalized k-reverse derivation of Γ-ring, and γ-homomorphism, anti-γ-homomorphism of M. Also, we give some commutattivity conditions on γ-prime Γ-ring and γ-semiprime Γ-ring .
Let R be a commutative ring with identity and let Mbe a unitary R-module. We shall say that a proper submodule N of M is nearly S-primary (for short NS-primary), if whenever , , with implies that either or there exists a positive integer n, such that , where is the Jacobson radical of M. In this paper we give some new results of NS-primary submodule. Moreover some characterizations of these classes of submodules are obtained.
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 be a commutative ring with identity , and be a unitary (left) R-module. A proper submodule of is said to be quasi- small prime submodule , if whenever with and , then either or . In this paper ,we give a comprehensive study of quasi- small prime submodules.
Let be a unitary left R-module on associative ring with identity. A submodule of is called -annihilator small if , where is a submodule of , implies that ann( )=0, where ann( ) indicates annihilator of in . In this paper, we introduce the concepts of -annihilator-coessential and - annihilator - coclosed submodules. We give many properties related with these types of submodules.
The present work reports on the performance of three types of nanofiltration membranes in the removal of highly polluting and toxic lead (Pb2+) and cadmium (Cd2+) from single and binary salt aqueous solutions simulating real wastewaters. The effect of the operating variables (pH (5.5-6.5), types of NF membrane and initial ions concentration (10-250 ppm)) on the separation process and water flux was investigated. It was observed that the rejection efficiency increased with increasing pH of solution and decreasing the initial metal ions concentrations. While the flux decreased with increasing pH of solution and increasing initial metal ions concentrations. The maximum rejection of lead and cadmium ion
... Show MoreThe paper comprise comparative palynological study of six species belong to the genera Urtica L. and Parietaria L.(P.alsinifolia Del., P.lusitanica L., P.judaicaL., U.urens L., U.dioica L., and U.pilulifera L.) of the Family Urticaceae in Iraq. All pollen grains were small size and found to be porate, Stephanoporate, Zonoporate. Characters such as shape of pollen grain, number and Dimention of pores, and Sculpturing of pollen grains were overlapped between species and of limited taxonomic value.
We introduce in this paper the concept of approximaitly semi-prime submodules of unitary left -module over a commutative ring with identity as a generalization of a prime submodules and semi-prime submodules, also generalization of quasi-prime submodules and approximaitly prime submodules. Various basic properties of an approximaitly semi-prime submodules are discussed, where a proper submodule of an -module is called an approximaitly semi-prime submodule of , if whenever , where , and , implies that . Furthermore the behaviors of approximaitly semi-prime submodule in some classes of modules are studied. On the other hand several characterizations of this concept are
... Show MoreEndometriosis, an autoimmune disease is among one of the most challenging of the 21 st century that affects women in reproductive age. The aim of the present study is to highlight the role of IL-36γ, and its relationship with other cytokines (Ang-2 and TNF-α) in the pathogenesis of endometriosis . Seventy five (75) consecutive women patient of reproductive age (25-40) years were enrolled in this study , Patients were divided into three groups , group 1 (G1) included (25) newly diagnosed endometriotic patients , that were not given any treatment related to Gynecology or anti-inflammatory medications . Second group : group 2 (G2) consists of (25) endometriotic patient who were treated with zoladex
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