The main objective of this thesis is to study new concepts (up to our knowledge) which are P-rational submodules, P-polyform and fully polyform modules. We studied a special type of rational submodule, called the P-rational submodule. A submodule N of an R-module M is called P-rational (Simply, N≤_prM), if N is pure and Hom_R (M/N,E(M))=0 where E(M) is the injective hull of M. Many properties of the P-rational submodules were investigated, and various characteristics were given and discussed that are analogous to the results which are known in the concept of the rational submodule. We used a P-rational submodule to define a P-polyform module which is contained properly in the polyform module. An R-module M is called P-polyform if every essential submodule of M is P-rational in M. We study this kind of module in some detail and introduced some characterizations of the P-polyform module and its relationships with some other modules. The third kind of module in this thesis is called fully polyform module, and it is contained in the class of polyform module. A module M is said to be fully polyform, if every P-essential submodule of M is rational in M, that is Hom_R(M/N, E(M))=0 for every P-essential submodule N of M. In fact, the class of fully polyform modules lies between polyform modules and essentially quasi-Dedekind modules. The main characteristics of fully polyform modules were investigated, and some characterizations of these types of modules were established. Furthermore, the relationships between this class and other related modules were examined.
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Let R be a ring and let M be a left R-module. In this paper introduce a small pointwise M-projective module as generalization of small M- projective module, also introduce the notation of small pointwise projective cover and study their basic properties.
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The concept of epiform modules is a dual of the notion of monoform modules. In this work we give some properties of this class of modules. Also, we give conditions under which every hollow (copolyform) module is epiform.
The purpose of this paper is to study a new types of compactness in the dual bitopological spaces. We shall introduce the concepts of L-pre- compactness and L-semi-P- compactness .
Let f and g be a self – maps of a rational exterior space . A natural number m is called a minimal coincidence period of maps f and g if f^m and g^m have a coincidence point which is not coincidence by any earlier iterates. This paper presents a complete description of the set of algebraic coincidence periods for self - maps of a rational exterior space which has rank 2 .
A new generalizations of coretractable modules are introduced where a module is called t-essentially (weakly t-essentially) coretractable if for all proper submodule of , there exists f End( ), f( )=0 and Imf tes (Im f + tes ). Some basic properties are studied and many relationships between these classes and other related one are presented.
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Expanded use of antibiotics may increase the ability of pathogenic bacteria to develop antimicrobial resistance. Greater attention must be paid to applying more sustainable techniques for treating wastewater contaminated with antibiotics. Semiconductor photocatalytic processes have proven to be the most effective methods for the degradation of antibiotics. Thus, constructing durable and highly active photocatalytic hybrid materials for the photodegradation of antibiotic pollutants is challenging. Herein, FeTiO3/Fe-doped g-C3N4 (FTO/FCN) heterojunctions were designed with different FTO to FCN ratios by matching the energy level of semiconductors, thereby developing effective direct Z-type heterojunctions. The photodegradation behaviors of th
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