Let R be a commutative ring with non-zero identity element. For two fixed positive integers m and n. A right R-module M is called fully (m,n) -stable relative to ideal A of , if for each n-generated submodule of Mm and R-homomorphism . In this paper we give some characterization theorems and properties of fully (m,n) -stable modules relative to an ideal A of . which generalize the results of fully stable modules relative to an ideal A of R.

In this paper ,we introduce hollow modules with respect to an arbitrary submodule .Let M be a non-zero module and T be a submodule of M .We say that M is aT-hollow module if every proper submodule K of M such that T ⊈ K is a T-small submodule of M .We investigate the basic properties of a T-hollow module

New types of modules named Fully Small Dual Stable Modules and Principally Small Dual Stable are studied and investigated. Both concepts are generalizations of Fully Dual Stable Modules and Principally Dual Stable Modules respectively. Our new concepts coincide when the module is Small Quasi-Projective, and by considering other kind of conditions. Characterizations and relations of these concepts and the concept of Small Duo Modules are investigated, where every fully small dual stable R-module M is small duo and the same for principally small dual stable.

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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    Let 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,) there exists a submodule X of  such that  f (N)  X ≈ M, where  is the quasi-injective hull of M. And we call M N-quasi-tight, if every quotient N / K of N which embeds in  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

The present study introduces the concept of J-pure submodules as a generalization of pure submodules. We  study some of its basic  properties  and  by using this concept we  define the class of  J-regular modules,  where an R-module  M is called  J-regular module if every submodule of M is J-pure submodule. Many results about this concept are proved

  In this paper it was presented the idea quasi-fully cancellation fuzzy modules and we will denote it by  Q-FCF(M), condition universalistic idea quasi-fully cancellation modules It .has been circulated to this idea quasi-max fully cancellation fuzzy modules and we will denote it by Q-MFCF(M). Lot of results and properties have been studied in this research.