An Approach to Generalized Extending Modules Via Ec-Closed Submodules

In this article, we introduce a class of modules that is analogous of generalized extending modules. First we define a module M to be a generalized ECS if and only if for each ec-closed submodule A of M , there exists a direct summand D of M such that 𝐷 𝐴 is singular


Introduction.
Throughout this paper, all rings R are associative with unitary and all modules are unital right R-modules.In [1], for a module M, we say that M is generalized extending module if for each submodule A of M, there is a direct summand D of M such that   is singular.Obviously, every extending module is generalized extending.In [2], Kamal and Elmnophy introduced the concept of ec-closed submodules as a closed submodule which contains essentially cyclic submodule.Note that every direct summand of ec-closed submodule of M is ec-closed.As a generalization of CS-modules, a module M is called ECS if every ec-closed submodule of M is a direct summand of M, see [3].It is well known that von Neumann regular rings are ECS-modules, see [3] is singular, see [4].
In this paper, we study a module including the condition of generalized extending on the set of all ec-closed submodules of a module.We call a module M is generalized ECS if for every ec-closed submodule A of M, there is a direct summand D of M such that  ≤ . .A ring R is generalized ECS if   is generalized ECS R-module.The notion of generalized ECS property contains the class of generalized extending modules and ECS modules.
In section 2, we present basic properties of c-singular submodules with examples.Also, we consider connections between generalized ECS property, ECS and generalized extending conditions with examples.Moreover, we give sufficient conditions under which generalized ECS and ECS modules are equivalent.Also some characterizations of generalized ECS modules are given in this section.
In Section 3, we show by an example that the direct sum of generalized ECS modules need not be generalized ECS.We focus when a direct sum of generalized ECS modules is also generalized ECS.

Preliminary Results.
In this section, we recall c-singular submodules and obtain some properties of these submodules, also we use these submodules to introduce a generalization of ECS modules.
The generalized ECS notion is based on two tools, namely an ec-closed submodules of a module M and c-singular submodules.Let us begin by mentioning basic facts about them.

Examples and Remarks 2.1:
(1) If M is singular module, then every submodule of M is c-singular, for example is singular for every submodule A of   , hence every submodule of Zn as Z-module is c-singular.
(2) If M is uniform module, then every submodule of M is c-singular.The converse is not true in general, for example, Z6 as Z-module.
(3) Let M be an R-module and let A and B be submodules of M with A≤B, if A≤e B, then A≤c.s B. The converse is true when M is nonsingular.
Recall that a submodule Now, we study the basic properties of c-singular submodules.

Decompositions
It is well known that the direct sum of singular modules is also singular.But a direct sum of ECS modules may not be ECS.Also, a direct sum of generalized ECS modules need not be generalized ECS.
There are non-singular modules  =  1 ⨁ 2 in which  1 and  2 are ECS, but M is not ECS.(For example, Let R = [] be a polynomial ring of integers and let M = []⨁[]).Note that [] is an ECS, by [8, P.109] and hence generalized ECS but M is not ECS which is nonsingular, thus by proposition 2.11 M is not generalized ECS.
Next, we give various conditions under which the direct sum of generalized ECS is generalized ECS.
By similar argument, one can easily prove the following propositions.Proposition 3.2: Let M = M1  M2 be a duo module if M1 and M2 are generalized ECS modules, then M is generalized ECS.Proposition 3.3: Let M1 and M2 be generalized ECS modules such that annM1+annM2 = R, then M1  M2 is generalized ECS.Proposition 3.4: Let M = M1  M2 be an R-module with M1 being generalized ECS and M2 is semisimple.Suppose that for any ec-closed submodule A of M, A∩M1 is a direct summand of A, then M is generalized ECS.Proof: Let A be an ec-closed submodule of M, then it is easy to see that A+M1 = M  [(A+M1)∩M2].Since M2 is semisimple, then (A+M1)∩M2 is a direct summand of M2 and therefore A+M1 is a direct summand of M. By our assumption, A∩M1 is a direct summand of A, then A = (A∩M1)  A', for some submodule A' of A. It is clear that A∩M1 is ec-closed in M1.But M1 is generalized ECS, therefore, there is a direct summand D of M1 such that (A∩M1) ≤ . D and hence A = ((A∩M1)  A') ≤ . ⨁′.Since ⨁′ ≤ ⨁  +  1 ≤ ⨁ .Thus, M is generalized ECS.Proposition 3.5: Let M = M1  M2 such that M1 is generalized ECS, and M2 is an injective module.Then M is a generalized ECS if and only if for every ec-closed submodule A of M such that A∩M2≠0, there is a direct summand D of M such that  ≤ . .Proof: Suppose that for every ec-closed submodule A of M such that A∩M2≠0 there is a direct summand D of M such that  ≤ . .Let A be an ec-closed submodule of M such that A∩M2 = 0.By [8, Lemma 7.6, P. 57], there is a submodule ≅M1 is a generalized ECS and by by Proposition 2.2 we have A is an ecclosed submodule of M', we obtain there is a direct summand K of M', and hence of M such that A≤ . K. Thus, M is generalized ECS.The converse is obvious.
The following example shows that there is a submodule of generalized ECS which is not generalized ECS: Example 3.6: Let  = (   0  ).Since R is an indecomposable such that R is not extending, then R is not ECS as R is non-singular.By proposition 2.4 we have R is not a generalized ECS.But as   is a submodule of its injective hull   , which is an extending module then R is a generalized ECS." Under certain conditions the submodules of generalized ECS module may be also generalized ECS.

Proposition 3.7:
Let M be a generalized ECS and let A be a closed submodule of M such that the intersection of A with any direct summand of M is a direct summand of A, then A is generalized ECS.Proof: Let A be a closed submodule of M such that the intersection of A with any direct summand of M is a direct summand of A and let X be an ec-closed submodule of A. By Lemma 2.8, we have X is an ec-closed in M.But M is generalized ECS, there is a direct summand D of M such that X≤ . , hence X≤ .  ∩ .By our assumption, we get  ∩  is a direct summand of A. Thus, A is generalized ECS.

Corollary 3.8:
Let A be a direct summand of a generalized ECS module M such that the intersection of A with any direct summand of M is a direct summand of A, then A is generalized ECS.Corollary 3.9: Let M be a generalized ECS with the SIP, then every direct summand of M is a generalized ECS.

Proposition 2 . 5 : 6 :Proposition 2 . 7 : 2 . 8 :.
Let A ≤ B ≤ M such that A≤c.s M, then B≤c.s M. Proof: Let f : , f (m+A) m+B,  m M. It is clear that that f is epimorphismLet A and B be submodules of an R-module M. If A≤c.s M, then A+B≤c.sM. Let f : M → M' be a homomorphism and let A be a submodule of M such that A≤c.s M, then f(A)≤c.sf(M).Proof: Let A be a submodule of M with A≤c.s M. By First and Third Isomorphism theorems we have, up some basic properties of ec-closed submodule.Lemma Let M be an R-module, then the following are hold.(i) M has an ec-closed submodule.(ii)If A is an ec-closed submodule of M, and B is an essential submodule of M, then A∩B is ec-closed in B. (iii) Let A≤B≤M, if A is an ec-closed in M, then A is an ec-closed in B. (iv) Let {  } and {  } be collections of submodules of M, if   ec-closed submodules of   , for each α, then ⨁  is ec-closed in ⨁  .(v) If A is ec-closed in M, then A B ≤e A M , whenever B≤e M with A ≤ B. (vi) If A is an ec-closed in B and B is closed in M, then A is ec-closed in M. Proof:Theorem 2.13: Let M be an R-module.Then the following statements are equivalent.(i) M is generalized ECS module.(ii)For every ec-closed submodule A of M, there is a decomposition M = D  D', such that (D'+A) ≤ . M. (iii) For every ec-closed submodule A of M, there is a decomposition is a direct summand of M and K≤ . M. Proof: (i)⟹(ii) Let M be a generalized ECS and let A be an ec-closed submodule of M. Since there is a direct summand D of M such that A ≤ . D, then M = D  D', D' ≤ M. Since  ∩  ′ ≤  ∩  ′ = 0, then {A, D'} is an independent family, hence (A+D') ≤ . M, by Proposition 2.4.(ii)⟹(iii)Let A be an ec-closed submodule of M. By (ii), there is a decomposition M = D  D', such that (D'+A) ≤ . M. Claim that By taking K = D'+A and L = D, so we obtain the result.(iii)⟹(i) To show that M is generalized ECS, let A be an ec-closed submodule of M. By (iii), there is a decomposition L is a direct summand of M and K≤ . M. It is enough to show that A≤ . L. Let i :L→M be the injection map.Since K≤ . M, then i -1 (K)