‎Throughout this paper‎,  will denote a commutative ring with‎ ‎identity and  will denote the ring of integers.Let be an -module‎. A submodule  of is said to be pure if for every ideal of .  has the copure sum property if the sum of any two copure submodules is again copure‎.  is said to be a comultiplication module if for every submodule of  there exists an ideal  of such that .  satisfies the double annihilator conditions if for each ideal  of , we have . is said to be a strong comultiplication module if  is a comultiplication R-module which satisfies the double annihilator conditions. A submodule  of  is called fully invariant if for every endomorphism In [5]‎, ‎H‎. ‎Ansari-Toroghy and F‎. ‎Farshadifar introduced the dual notion of pure submodules (that is copure submodules) and investigated the first properties of this class of modules‎. ‎A submodule  of  is said to be copure if  for every ideal of .

We say that an -modulehas the copure intersection property if the intersection of any two copure submodules is again copure‎. In this paper, we investigate the modules with the copure intersection property and obtain some related results.

The following conclusions were drawn from this research.Every distributive -module has the copure intersection property.Every strong comultiplication -module has the copure intersection property.An -module  has the copure intersection property if and only if for each ideal  of and copure submodules  of  we have If  is a , then an -module  has the copure intersection property if and only if  has the copure sum property. Let , where is a submodule of . If  has the copure intersection property, then each  has the has the copure intersection property. The converse is true if each copure submodule of  is fully invariant.

In this paper, we investigate radical Noetherian modules as a collection of modules whose lattice of radical submodules is Noetherian. The collection of radical Noetherian modules contains both families of Noetherian and Artinian modules properly. We will show that the set of minimal prime submodules of a radical Noetherian modules is finite. Also a ring $R$ is called radical Noetherian, if $R$ is a radical Noetherian $R$-module. We will prove that a multiplication $R$-module $M$ is radical Noetherian if and only if $R/Ann(M)$ is a radical Noetherian. Moreover, we will give and prove analogs of Cohen and Hilbert basis theorems for radical Noetherian rings.

Throughout this paper R will denote an associative ring with identity, M a unitary right R-module. A functor 𝜏from the category of the right R-modules Mod-R to itself is called a preradicalif it satisfies the following properties: (i) 𝜏(𝑀)is a submodule of M, for every R-module M; (ii) If 𝑓: 𝑀′ → 𝑀is an R-module homomorphism, then 𝑓(𝜏(𝑀′ )) ≤ 𝜏(𝑀) and 𝜏(𝑓) is the restriction of 𝑓to 𝜏(𝑀′ ). For example Rad, Soc, and 𝑍𝑀are preradicals. Note that if K is a summand of M, then 𝐾 ∩ 𝜏(𝑀) = 𝜏(𝐾). For a preradical 𝜏, Al-Takhman, Lomp and Wisbauer defined and studied the concept of 𝜏-lifting and 𝜏-supplemented modules. A module M is called 𝜏- lifting if every submodule N of M has a decomposition 𝑁 = 𝐴 ⊕ 𝐵 such that A is a direct summand of M and 𝐵 ⊆ 𝜏(𝑀).A submodule 𝐾 ⊆ 𝑀 is called 𝜏 −supplement (weak𝜏-supplement) provided there exists some 𝑈 ⊆ 𝑀such that 𝑀 = 𝑈 + 𝐾 and 𝑈 ∩ 𝐾 ⊆ 𝜏(𝐾) (𝑈 ∩ 𝐾 ⊆ 𝜏(𝑀)). M is called 𝜏-supplemented (weakly 𝜏-supplemented) if each of its submodules 𝜏-supplement (weak 𝜏-supplement) in M.Talebi, Moniri Hamzekolaei and Keskin-Tütüncü, defined 𝜏-H-supplemented modules. A module M is called𝜏- H-supplemented if for every 𝑁 ≤ 𝑀 there exists a direct summand D of Msuch that (𝑁 + 𝐷)/𝑁 ⊆ 𝜏(𝑀/𝑁)and(𝑁 + 𝐷)/𝐷 ⊆ 𝜏(𝑀/𝐷). The 𝛽 ∗ relation is introduced and investigated by Birkenmeier, Takil Mutlu, Nebiyev, Sokmez and Tercan. Let X and Y be submodules of M. X and Yare 𝛽 ∗ equivalent, X𝛽 ∗Y, provided 𝑋+𝑌 𝑋 ≪ 𝑀 𝑋 𝑎𝑛𝑑 𝑋+𝑌 𝑌 ≪ 𝑀 𝑌 . Based on definition of 𝛽 ∗ relation they introduced two new classes of modules namely 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ -lifting and 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ −supplemented.They showed that two concept of H-supplemented modules and 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ −lifting modules coincide. In this paper, we introduce Goldie−𝜏 −supplemented and strongly 𝜏-Hsupplemented modules. We introduce the̅𝛽̅̅̅∗ relation. We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie−𝜏 −supplemented and strongly 𝜏-H-supplemented modules. We call a module M, Goldie−𝜏 −supplemented (strongly 𝜏-H-supplemented) if for any submodule N of M, there exists a 𝜏-supplement submodule (a direct summand) D of M such thatN𝛽̅̅̅∗D. Clearly every strongly 𝜏-H-supplemented module is Goldie 𝜏 -supplemented. We will study direct sums of Goldie 𝜏 -Hsupplemented modules. Let 𝑀 = 𝐴 ⊕ 𝐵 be a distributive module. Then M is Goldie 𝜏 -upplemented (strongly 𝜏 -H-supplemented) if and only if A and B are Goldie 𝜏 -supplemented (strongly 𝜏 -H-supplemented). We also define 𝜏 -Hcofinitely supplemented modules and obtain some conditions which under the factor module of a 𝜏 -H-cofinitely supplemented module will be 𝜏 -H-cofinitely supplemented.

In this paper, first we define and investigate the 𝛽𝜏 ∗ relation on submodules of a module. We show that the 𝛽𝜏 ∗ relation is an equivalence relation. We apply this relation to define and investigate the classes of Goldie-𝜏 -supplemented modules and strongly𝜏-H-supplemented modules.

We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie−𝜏 −supplemented and strongly 𝜏-Hsupplemented modules. We call a module M, Goldie−𝜏 −supplemented (strongly 𝜏 -H-supplemented) if for any submodule N of M, there exists a 𝜏- supplement submodule (a direct summand) D of M such that N𝛽̅̅̅∗ D. Clearly every strongly 𝜏 -H-supplemented module is Goldie 𝜏 -supplemented. We will study direct sums of Goldie 𝜏 -H-supplemented modules. Let 𝑀 = 𝐴 ⊕ 𝐵 be a distributive module. Then M is Goldie 𝜏 -upplemented (strongly 𝜏 -Hsupplemented) if and only if A and B are Goldie 𝜏 -supplemented (strongly 𝜏 - H-supplemented). We also define 𝜏 -H-cofinitely supplemented modules and obtain some conditions which under the factor module of a 𝜏 -H-cofinitely supplemented module will be 𝜏 -H-cofinitely supplemented.

The following conclusions were drawn from this research.  Let 𝑀 = 𝑀1 ⊕ 𝑀2 , where 𝑀1 is a fully invariant submodule of M. Assume that 𝜏 is a cohereditary preradical. If M is strongly 𝜏-Hsupplemented, then 𝑀1 𝑎𝑛𝑑 𝑀2 are strongly 𝜏 −H-supplemented.  Let M be an 𝜏-H-cofinitely supplemented module and let 𝑁 ≤ 𝑀 be a submodule. Suppose that for every direct summand K of M, there exists a submodule L of M such that 𝑁 ⊆ 𝐿 ⊆ 𝐾 + 𝑁, L/N is a direct summand of M/N and 𝐾+𝑁 𝑁 𝐿/𝑁 ⊆ 𝜏( 𝑀 𝑁 )+ 𝐿 𝑁 𝐿/𝑁 . Then M/N is 𝜏-Hcofinitelysupplemented.  Let M be a module and let 𝑁 ≤ 𝑀 be a submodule such that for each decomposition 𝑀 = 𝑀1 ⊕ 𝑀2 we have 𝑁 = (𝑁 ∩ 𝑀1 ) ⊕ (𝑁 ∩ 𝑀2). If M is 𝜏-H-cofinitely supplemented, then M/N is 𝜏-H-cofinitely supplemented.

An R-module M is called Almost uniserial module, if any two non-isomorphic submodules of M are linearly ordered by inclusion. In this paper, we investigate some properties of Almost uniserial modules. We show that every finitely generated Almost uniserial module over a Noetherian ring, is torsion or torsionfree. Also the construction of a torsion Almost uniserial modules whose first nonzero Fitting ideal is a product of maximal ideals, is invetigated and torsion Almost uniserial modules over an integral domain and a UFD are characterized.

Throughout this paper,  is a commutative ring with non-zero identity and  is an  -module. The study of injective modules is very important in commutative algebra and homological Algebra. Any product of (even finitely many) injective modules is injective; conversly, if a direct product of modules is injective, then each module is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sum of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple. Also every factor module of every injective module is injective if and only if the ring is Hereditary. Finally every infinite direct sum of injective modules is injective if and only if the ring is Noetherian. In this paper we study some new  propertis of this modules.

The main tool used in the proofs of the main results of this paper is the properties of injective modules and injective envelopes.

We present some new properties of injective envelopes, injective modules, prime submodules and maximal submodules.

We prove the following

Over finitely generated multiplication modules, every prime submodule is irreducible.If  N is a prime submodule of finitely generated multiplication R-module M such that E(M/N) is finitely generated, then N is a maximal submodule of M. Also we give several  corollaries for this note. Also we find relations beetwen Artinian ring, Noetherian ring, indecomposable injective modules and injective cogenerators of  modules.

Let R be an arbitrary ring and N be a right R-module.  A right R-module M is called N-retractable if HomR(M,N')≠0, for any nonzero submodule N' of N. This is a generalization of the concept of retractable modules. The aim of this paper is to study of N-retractable modules, where N is an arbitrary right R-module. One of the most important results of this paper is the characterization of rings that have a module such that each module is retractable with respect to it. Also we show that the class of N-retractable modules is closed under direct sums and direct products.

Throughout this paper,  is a commutative Noetherian ring with non-zero identity,  is an ideal of ,  is a finitely generated -module, ‎and  is an arbitrary -module which is not necessarily finitely generated. Let L be a finitely generated R-module. Grothendieck, in [11], conjectured that  is finitely generated for all . In [12], ‎Hartshorne gave a counter-example and raised the question whether  is finitely generated for all  and . The th generalized local cohomology module of  and  with respect to ,was introduced by Herzog in [14]. It is clear that  is just the ordinary local cohomology module  of  with respect to . As a generalization of Hartshornechr('39')s question, we have the following question for generalized local cohomology modules (see [25, Question 2.7]).Question. When is  finitely generated for all  and ? In this paper, we study  in general for a finitely generated -module  and an arbitrary -module .

The main tool used in the proofs of the main results of this paper is the spectral sequences.

We present some technical results (Lemma 2.1 and Theorems 2.2, 2.9, and 2.14) which show that, in certain situation, for non-negative integers , , , and  with ,  and the -modules  and  are in a Serre subcategory of the category of -modules (i.e. the class of   -modules which is closed under taking submodules, quotients, and extensions).

We apply the main results of this paper to some Serre subcategories (e.g. the class of zero  -modules and the class of finitely generated -modules) and deduce some properties of generalized local cohomology modules. In Corollaries 3.1-3.3, we present some upper bounds for the injective dimension and the Bass numbers of generalized local cohomology modules. We study cofiniteness and minimaxness of generalized local cohomology modules in Corollaries 3.4-3.8. Recall that, an -module  is said to be -cofinite (resp. minimax) if  and  is finitely generated for all  [12] (resp. there is a finitely generated submodule  of  such that  is Artinian [26]) where We show that if  is finitely generated for all  and  is minimax for all , then  is -cofinite for all  and  is finitely generated (Corollary 3.5). We prove that if  is finitely generated for all , where  is the arithmetic rank of , and  is -cofinite for all , then  is also an -cofinite -module (Corollary 3.6). We show that if  is local, , and  is finitely generated for all , then  is -cofinite for all  if and only if  is finitely generated for all  (Corollary 3.7). We also prove that if  is local, ,  is finitely generated for all , and  (or ) is -cofinite for all , then  is -cofinite for all  (Corollary 3.8). In Corollary 3.9, we state the weakest possible conditions which yield the finiteness of associated prime ideals of generalized local cohomology modules. Note that, one can apply the main results of this paper to other Serre subcategories to deduce more properties of generalized local cohomology modules../files/site1/files/71/15.pdf