Let be a ring with identity. Recall that a submodule of a left -module is called strongly essential if for any nonzero subset of , there is such that , i.e., . This paper introduces a class of submodules called se-closed, where a submodule of is called se-closed if it has no proper strongly essential extensions inside . We show by an example that the intersection of two se-closed submodules may not be se-closed. We say that a module is have the se-Closed Intersection Property, briefly se-CIP, if the intersection of every two se-closed submodules of is again se-closed in . Several characterizations are introduced and studied for each of these concepts. We prove for submodules and of that a module has the se-CIP if and only if is strongly essential in implies is strongly essential in . Also, we verify that, a module has the se-CIP if and only if for each se-closed submodule of and for all submodule of , is se-closed in . Finally, some connections and examples are included about (se-CIP)-modules
Let be a non-zero right module over a ring with identity. The weakly second submodules is studied in this paper. A non-zero submodule of is weakly second Submodule when , where , and is a submodule of implies either or . Some connections between these modules and other related modules are investigated and number of conclusions and characterizations are gained.
Let be a commutative ring with identity, and a fixed ideal of and be an unitary -module. In this paper we introduce and study the concept of -nearly prime submodules as genrealizations of nearly prime and we investigate some properties of this class of submodules. Also, some characterizations of -nearly prime submodules will be given.
There are two (non-equivalent) generalizations of Von Neuman regular rings to modules; one in the sense of Zelmanowize which is elementwise generalization, and the other in the sense of Fieldhowse. In this work, we introduced and studied the approximately regular modules, as well as many properties and characterizations are considered, also we study the relation between them by using approximately pointwise-projective modules.
Let be a ring with identity and be a submodule of a left - module . A submodule of is called - small in denoted by , in case for any submodule of , implies . Submodule of is called semi -T- small in , denoted by , provided for submodule of , implies that . We studied this concept which is a generalization of the small submodules and obtained some related results
In this research, we introduce a small essentially quasi−Dedekind R-module to generalize the term of an essentially quasi.−Dedekind R-module. We also give some of the basic properties and a number of examples that illustrate these properties.
Let R be a commutative ring with identity, and let M be a unity R-module. M is called a bounded R-module provided that there exists an element x?M such that annR(M) = annR(x). As a generalization of this concept, a concept of semi-bounded module has been introduced as follows: M is called a semi-bounded if there exists an element x?M such that . In this paper, some properties and characterizations of semi-bounded modules are given. Also, various basic results about semi-bounded modules are considered. Moreover, some relations between semi-bounded modules and other types of modules are considered.
Let R be a commutative ring with identity. R is said to be P.P ring if every principle ideal of R is projective. Endo proved that R is P.P ring if and only if Rp is an integral domain for each prime ideal P of R and the total quotient ring Rs of R is regular. Also he proved that R is a semi-hereditary ring if and only if Rp is a valuation domain for each prime ideal P of R and the total quotient Rs of R is regular. , and we study some of properties of these modules. In this paper we study analogue of these results in C.F, C.P, F.G.F, F.G.P R-modules.
Background: Extensive reports denote the biological role of trace elements (Cu Zn ", Se ') in many various physiological and pathological conditions, one of the list is shown to be Behcet's disease (BD).The aim of this study was directed towards the distribution of trace elements (Cu 2,Zn, 2,Se 2) in sera of BD patients, and control groups.
Methods: Fifty Arab Iraqi patients who fulfdled the ISO criteria for diagnosis BD.compared with 21 patients control with recurrent oral ulcer (ROU), healthy control groups (2Ihealthy relative and 21healthy volunteer). Typing HLA Class I was carried out at Alkarama hospital, by using microlympho-cytotoxicity test. The concentrations of serum Cu ', Zn ", and Se" were determined by atomic absorption s
Scientists are investigating the efficacy of different biosorbents for promoting economic and environmental viability in purifying contaminants. Among the primary by-products of biodiesel production is waste microalgae biomass, which has the potential to be used as a cheap biosorbent for the treatment of pollution. In the present study, the biomass left over after extracting the chlorella vulgaris was used to test the potential biosorption of CIP from simulated aqueous solutions. Bisorbent's ability was characterized using Fourier transform infrared spectroscopy (FTIR), scanning electron microscopy (SEM), and energy-dispersive X-ray spectroscopy (EDX). Analysis with a Fourier Transform Infrared Spectrometer revealed that C
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