The goal of this discussion is to study the twigged of pure-small (pr-small) sub- moduleof a module W as recirculation of a small sub-module, and we give some basic idiosyncrasy and instances of this kind of sub-module. Also, we give the acquaint of pure radical of a module W (pr-radical) with peculiarities.

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.

Let R be a Г-ring, and σ, τ be two automorphisms of R. An additive mapping d from a Γ-ring R into itself is called a (σ,τ)-derivation on R if d(aαb) = d(a)α σ(b) + τ(a)αd(b), holds for all a,b ∈R and α∈Γ. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)]_α = [a,b]_α^(σ,τ) holds for all a,b∈R and α∈Γ. In this paper, we investigate the commutativity of R by the strong commutativity preserving (σ,τ)-derivation d satisfied some properties, when R is prime and semi prime Г-ring.

   Let  be a commutative ring with identity, and  be a unitary left R-module. In this paper we, introduce and study a new class of modules called pure hollow (Pr-hollow) and pure-lifting (Pr-lifting). We give a fundamental, properties of these concept.  also, we, introduce some conditions under which the quotient and direct sum of Pr-lifting modules is Pr-lifting.

        Throughout this paper, three concepts are introduced namely stable semisimple modules, stable t-semisimple modules and strongly stable t-semisimple. Many features co-related with these concepts are presented. Also many connections between these concepts are given. Moreover several relationships between these classes of modules and other co-related classes and other related concepts are introduced.

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

Let R be a commutative ring with identity and M be unitary (left) R-module. The principal aim of this paper is to study the relationships between relatively cancellation module and multiplication modules, pure submodules and Noetherian (Artinian) modules.

The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .

Suppose R has been an identity-preserving commutative ring, and suppose V has been a legitimate submodule of R-module W. A submodule V has been J-Prime Occasionally as well as occasionally based on what’s needed, it has been acceptable: x ∈ V + J(W) according to some of that r ∈ R, x ∈ W and J(W) an interpretation of the Jacobson radical of W, which x ∈ V or r ∈ [V: W] = {s ∈ R; sW ⊆ V}. To that end, we investigate the notion of J-Prime submodules and characterize some of the attributes of has been classification of submodules.