Jordan  curve  theorem  is  one  of  the  classical  theorems  of  mathematics, it states  the  following :  If    is a graph of  a  simple  closed curve  in  the complex plane the complement  of   is the union of  two regions,  being the common  boundary of the two regions. One of  the region   is  bounded and the other is unbounded. We introduced in this paper one of Jordan's theorem generalizations. A new type of space is discussed with some properties and new examples. This new space called Contractible -space.

Let  be a commutative ring with an identity and be a unitary -module. We say that a non-zero submodule  of  is  primary if for each with en either or  and an -module  is a small primary if   =  for each proper submodule  small in. We provided and demonstrated some of the characterizations and features of these types of submodules (modules).  

Let  be a commutative ring with identity and let   be an R-module. We call an R-submodule  of  as P-essential if  for each nonzero prime submodule  of    and 0  . Also, we call an R-module  as P-uniform if every non-zero submodule  of  is P-essential. We give some properties of P-essential and introduce many properties to P-uniform R-module. Also, we give conditions under which a submodule  of a multiplication R-module  becomes P-essential. Moreover, various properties of P-essential submodules are considered.

Let R be a commutative ring with unity and let M be a left R-module. We define a proper submodule N of M to be a weakly prime if whenever  r  R,  x  M, 0  r x  N implies  x  N  or  r  (N:M). In fact this concept is a generalization of the concept weakly  prime ideal, where a proper ideal P of R is called a weakly prime, if for all a, b  R, 0  a b  P implies a  P or b  P. Various properties of weakly prime submodules are considered. 

Let R be a ring with identity and let M be a left R-module. M is called µ-lifting modulei f for every sub module A of M, There exists a direct summand D of M such that M = D D', for some sub module D' of M such that A≤D and A D'<<_µ D'. The aim of this paper is to introduce properties of µ-lifting modules. Especially, we give characterizations of µ-lifting modules. On the other hand, the notion of amply µ-supplemented iis studied as a generalization of amply supplemented modules, we show that if M is amply µ-supplemented such that every µ-supplement sub module of M

The aim of this work is to give the new types for diskcyclic criterion. We also introduced the case if there is an equivalent relation between a diskcyclic operator  and T that satisfies the diskcyclic criterion. Moreover, we discussed the condition that makes T, which satisfies the diskcyclic criterion, a diskcyclic operator

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A gamma T_ pure sub-module also the intersection property for gamma T_pure sub-modules have been studied in this action. Different descriptions and discuss some ownership, as Γ-module Z owns the TΓ_pure intersection property if and only if (J2 ΓK ∩ J^2  ΓF)=J^2 Γ(K ∩ F) for each Γ-ideal J and for all TΓ_pure K, and F in Z Q/P is TΓ_pure sub-module in Z/P, if P in Q.