Let R be a ring with 1 and W is a left Module over R. A Submodule D of an R-Module W is small in W(D ≪ W) if whenever a Submodule V of W s.t W = D + V then V = W. A proper Submodule Y of an R-Module W is semismall in W(Y ≪_S W) if Y = 0 or Y/F ≪ W/F ∀ nonzero Submodules F of Y. A Submodule U of an R-Module E is essentially semismall(U ≪es E), if for every non zero semismall Submodule V of E, V∩U ≠ 0. An R-Module E is essentially semismall quasi-Dedekind(ESSQD) if Hom(E/W, E) = 0 ∀ W ≪es E. A ring R is ESSQD if R is an ESSQD R-Module. An R-Module E is a scalar R-Module if, ∀ , ∃ s.t V(e) = ze ∀ . In this paper, we study the relationship between ESSQD Modules with scalar and multiplication Modules. We show that if E is scalar semismall quasi-prime R-Module. Then E is an ESSQD R-Module, we show that if E is faithful multiplication R-Module, thus E is an essentially semismall prime R-Module iff R is an ESSQD ring
Let R be a commutative ring with unity. In this paper we introduce and study the concept of strongly essentially quasi-Dedekind module as a generalization of essentially quasiDedekind module. A unitary R-module M is called a strongly essentially quasi-Dedekind module if ( , ) 0 Hom M N M for all semiessential submodules N of M. Where a submodule N of an R-module M is called semiessential if , 0  pN for all nonzero prime submodules P of M .
Let R be a commutative ring with identity . In this paper we study the concepts of essentially quasi-invertible submodules and essentially quasi-Dedekind modules as a generalization of quasi-invertible submodules and quasi-Dedekind modules . Among the results that we obtain is the following : M is an essentially quasi-Dedekind module if and only if M is aK-nonsingular module,where a module M is K-nonsingular if, for each , Kerf ≤e M implies f = 0 .
Let R be a commutative ring with 1 and M be a (left) unitary R – module. This essay gives generalizations for the notions prime module and some concepts related to it. We termed an R – module M as semi-essentially prime if annR (M) = annR (N) for every non-zero semi-essential submodules N of M. Given some of their advantages characterizations and examples, and we study the relation between these and some classes of modules.
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.
An R-module M is called rationally extending if each submodule of M is rational in a direct summand of M. In this paper we study this class of modules which is contained in the class of extending modules, Also we consider the class of strongly quasi-monoform modules, an R-module M is called strongly quasi-monoform if every nonzero proper submodule of M is quasi-invertible relative to some direct summand of M. Conditions are investigated to identify between these classes. Several properties are considered for such modules
Suppose that A be an abelain ring with identity, B be a unitary (left) A-module, in this paper ,we introduce a type of modules ,namely Quasi-semiprime A-module, whenever is a Prime Ideal For proper submodule N of B,then B is called Quasi-semiprime module ,which is a Generalization of Quasi-Prime A-module,whenever annAN is a prime ideal for proper submodule N of B,then B is Quasi-prime module .A comprchensive study of these modules is given,and we study the Relationship between quasi-semiprime module and quasi-prime .We put the codition coprime over cosemiprime ring for the two cocept quasi-prime module and quasi-semiprime module are equavelant.and the cocept of prime module and quasi
... Show MoreLet R be a commutative ring with unity and let M, N be unitary R-modules. In this research, we give generalizations for the concepts: weakly relative injectivity, relative tightness and weakly injectivity of modules. We call M weakly N-quasi-injective, if for each f  Hom(N,ï) there exists a submodule X of ï such that f (N) ïƒ X ≈ M, where ï is the quasi-injective hull of M. And we call M N-quasi-tight, if every quotient N / K of N which embeds in ï embeds in M. While we call M weakly quasi-injective if M is weakly N-quasiinjective for every finitely generated R-module N. Moreover, we generalize some properties of weakly N-injectiv
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