Conseguences of soil crude oil pollution on some wood properties of olive trees


 

Mathematics |227 
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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

Modules with Chain Conditions on S-Closed Submodules 
 

Rana Noori Majeed Mohammed 

Dept. of Mathematic / College of Education for pure science-(Ibn Al-Haitham)/ 

University of Baghdad 

rana.n.m@ihcoedu.uobaghdad.edu.iq 

 

Received in :31 /May /2017, Accepted in : 27 /August/ 2017 

 

 

Abstract 
      Let L be a commutative ring with identity and let W be a unitary left L- module. A 

submodule D of an L- module W is called  s- closed submodule denoted by  D ≤sc W, if D has   

no  proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D 

= H. In  this  paper,  we study  modules which satisfies  the ascending chain  conditions 

(ACC) and descending chain conditions (DCC) on this kind of submodules. 

 

Keywords:  s-essential submodules, s-closed submodules , ascending and descending chain 

conditions. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

mailto:rana.n.m@ihcoedu.uobaghdad.edu.iq


 

Mathematics |228 
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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

Introduction  
     Throughout this paper, L represents a commutative ring with unity and W be a left unitary 

L- module. Its well known that “a submodule D of W is called small denoted by D<< W if 

and only if D + U = W implies U=W for each U submodule of W (U≤W)” [2], and “a 

submodule D of an L - module W is called an essential submodule of W and denoted by 

D≤eW if every non-zero submodule of W has non-zero intersection with D” [3], while “a 

submodule D of an L- module W is said to be a closed submodule of W if D has no proper 

essential extension inside W, that is if D≤e H≤ W then D=H” [3]. As a generalization of 

essential submodules , in [4] “Zhou and Zhang” introduced the concept of s- essential 

submodule , where “a submodule D of an L-module W is said to be an s -essential submodule 

of W denoted by D≤se W if D∩H=0 with H is a small submodule of W implies H= 0. “Mehdi 

Sadiq and Faten” in [1] introduced and studied the notion of s- closed submodules, “a 

submodule D of an L- module W is called s-closed submodule denoted by D≤sc W, if D has 

no proper  s- essential extension in W, that is , whenever D ≤ W such that D≤se H≤ W , then 

D= H. 

 This paper consists of two sections. In section one, we give some other properties and 

examples of s-essential   submodules and s- closed submodules. In section two, we study 

chain conditions (that is ascending and descending chain conditions)  on s-closed submodules. 

 

1. S-Essential   Submodules and S- Closed Submodules 
Definition 1 . 1 : [4] 

     A submodule D of an L-module W is said to be an s- essential submodule of W denoted by 

D ≤se W if D∩H= 0 with H is a small submodule of W implies H= 0. 

Remarks and examples 1 . 2 : 

1) Its clear that every essential submodule is an s- essential submodule, hence every 

submodule of Z -module Z,   
  ( where P is a prime number, nZ+) is s- essential. 

2) If W is an L- module such that (0) is the only small submodule then every submodule 
is s- essential submodule in W. 

In particular , for each submodule of semisimple module ( or free Z ˗module ) is s- essential. 

Hence its clear that every submodule of Z ˗module Z6 is s- essential, however they are not 

essential. Also every submodule of the Z˗ module  Z  Z  is s- essential submodule.  

3) Let A be a submodule of an L -module W, then there exists a closed submodule H of 
W such that A≤e H, it is clear by [3, Exc.13, p.20], hence A≤se H. 

4) In Z24 as Z- module, we have < ̅>, < ̅>, < ̅>, < ̅>,  <  ̅̅̅̅ >, and Z24 are s- essential 
submodules in Z24, but < ̅> is not since < ̅>∩< ̅> ={0} while < ̅>  0 is a small submodule 
in Z24. 

5) For a nonzero R-module W, W ≤se W.  
6) The two concepts essential and s- essential are coincide under the class of hollow 
modules, by[1, Remark (2.3)], where “an L -module W is called Hollow if every proper 

submodule of W is small”. [5] 

Proposition 1 . 3 : 

     Let W be an L-module and let S ≤se T ≤ M and S ≤se Tʹ ≤ W, then S ∩ Sʹ ≤se T ∩ Tʹ. 

Proof: 

     Let U << T ∩ Tʹ and (S ∩ Sʹ) ∩ U = (0), hence S ∩ ( Sʹ ∩ U ) = 0. 

But U << ( T ∩ Tʹ ) implies U << Tʹ and U << T.  

As Sʹ∩U U<< T, then Sʹ∩U<< T. But S ≤se T, hence Sʹ∩U= 0. 

It follows that U = 0 since S ≤se Tʹ and U << Tʹ. 
     The following result follows by Proposition 1.3 directly. 

 



 

Mathematics |229 
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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

Corollary 1 . 4 : 

     Let C, D be submodules of W such that C≤seW  and D≤seW. Then C∩D ≤se W,  [4, 

proposition 2.7(1)(b)] . 

Proposition 1 . 5 : 

     Let W=W1⊕W2, and let A = A1⊕A2 ≤se B1⊕B2, where B1 ≤ W1 and B2≤ W2. Then A1≤se 
B1 and A2≤se B2. 

Proof: 

      Suppose A1 is not an s-essential submodule in B1. So there exists a nonzero small 

submodule D1 in B1 such that A1∩D1=(0). 

Since D1⊕(0) is a small submodule in B1⊕B2 and (A1⊕A2) ∩ (D1⊕(0)) = (A1∩D1) ⊕ 
(A2∩(0)) = (0). 

Then A1⊕A2 is not an s-essential submodule in B1⊕B2 which is a contradiction. 
Thus A1 ≤se B1 and by the same way of proof that A2 ≤se B2. 

 

Proposition 1 . 6 : 

     Let W be a faithful  multiplication  finitely generated (denoted by FMFG)  L- module, and 

U a submodule of W. Then U ≤se W if and only if there exists an s -essential ideal E of L such 

that U = EW. 

Proof: 

() Let U≤se W . As W is a multiplication L- module, so U= EW for some E ≤ L. To prove 

that E ≤se L, assume J is a small ideal of L and E ∩ J = 0, hence ( E ∩ J )W = 0. Then by [6, 

Th. 1.6(i), p. 759] EW ∩ JW = 0, that is U ∩ JW = 0 .  

But by [8, prop.1.1.8] JW is a small submodule of W and U ≤se W, so JW = 0. Hence J= 0 ( 

since W is a faithful module ) . Thus E ≤se L. 

() To prove U ≤se W. Assume V is a small submodule of W, hence V = JW for some J << L. 

if U ∩ V = 0, then EW ∩ JW = 0 and so (E ∩ J) W = 0. Hence E ∩ J = 0 since W is faithful. 

Thus J= 0 because E ≤se L. It follows that V= 0 and U ≤se W. 

Theorem 1 . 7 : 

     Let W be a FMFG L- module. Then I≤se J≤ L if and only if IW≤se JW. 

Proof: 

() Let U be a small submodule in JW≤ W, so U≤ W. Thus U= KW for some K≤ L. As 

KW≤JW then K≤ J, by [6, Th.3.1] 

To prove K is a small submodule in J , let K+H = J, so KW + HW = JW. That is HW =JW 

(since KW = U which is a small submodule in JW). Hence HW =JW and so H=J, that is K is a 

small submodule in J. 

If IW ∩U = 0, then IW ∩ KW =0. Thus (I∩K)W = 0 , so I∩K=0 (since W is faithful 

multiplication). 

But I≤se J and K is a small submodule in J, hence K= 0. It follows U= 0, thus IW≤se JW. 

() If IW≤se JW to prove I≤se J≤ L. Let K be a small submodule of J. 

Assume I∩K = 0, then (I∩K) W = 0, so IW ∩ KW = 0. 

Let KW +H = JW. Since W is a multiplication module , thus H = CW. Hence KW + CW = 

JW. 

Since KW is a small submodule in JW , then CW = JW and hence C = J. Thus H = JW and 

KW is a small submodule of JW. 

Now, IW ∩ KW =0 and KW is a small submodule in JW implies KW = 0 (since IW≤se JW) 

and so K=0. It follows I≤se J. 

     Recall that , “a non-zero L-module W is called small -uniform (shortly, by s -uniform) if 

every nonzero submodule of W is s -essential. A ring L is called s-uniform if L is an s-

uniform L-module”. [9] 

 



 

Mathematics |230 
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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

Corollary 1 . 8 : 

     Let W be a FMFG L-module. Then W is s-uniform module  if  and  only  if  L is s-uniform 

ring. 

Definition 1 . 9 : [1] 

     A submodule D of an L -module W is called s-closed submodule denoted by D ≤sc W, if D 

has no proper s -essential extension in W, that  is , whenever D ≤W such that D ≤se K ≤ W, 

then D = K. An ideal E of L is called an s-closed, if its an s- closed submodule in L. Where 

every s- closed submodule in W is closed in W but the converse is not true. 

Examples 1 . 10 : 

1) In Z24 as a Z-module. Z24 and <  ̅ > are the only s-closed submodules while 
< ̅>,< ̅>,< ̅>,< ̅> and <  ̅̅̅̅ > are not because they have a proper s-essential submodule which 
is Z24. All submodules of Z24 have the following properties. 

 

A ≤ Z24 A<< Z24 A ≤se Z24 A ≤sc Z24 

< ̅>    

< ̅>    

< ̅>    

< ̅>    

< >    

< ̅>    

<  ̅̅̅̅ >    

Z24    

 

Similarly, < ̅>,< ̅> and < ̅> are not small submodules in < ̅> in Z24 but <  ̅̅̅̅ > is a small 
submodule in < ̅> and < ̅>∩<  ̅̅̅̅ >{0} thus < ̅> is an s-essential submodule in < ̅>, so it is 
not an s-closed submodule in < ̅>.  
2) If W is a simple module , then < ̅> and W are s- closed submodules. 
3) Let W be an L-module. If every submodule of W is s-closed (hence every submodule 
is closed), then W is semisimple module, however the converse is not true, for example in Z6 , 

Z6 is a Z- module is semisimple but the submodules < ̅>, < ̅>, < ̅> are not s- closed. 
Proposition 1. 11 : 

      Let W be an L-module such that the s-essential submodules satisfy transitive property. 

Then for each A ≤ W, there exists an s-closed submodule such that A ≤se H. 

Proof: 

     Let S={K≤W : A ≤se K}. V since A V. So by “ Zorn’s Lemma” S has a maximal 

element say H.  

To prove H is an s -closed submodule in W. Assume H ≤se D ≤ W. 

Since A ≤se H and H ≤se D, then A ≤se D (by transitive property), and so D  S.  

Hence H = D ( by maximality of L ). Thus H is an s-closed submodule. 
 

     The following proposition has been given in [1], we will mention it with its proof for the 

sake of completness. 

Proposition 1 . 12 :  

     Lea A be a submodule of B, and let B an s-closed submodule of W, then (B/A) is an          

s -closed submodule of (W/A). 

Proof: 

     Assume (B/A) ≤se (C/A) where (C/A) ≤ (W/A). Let π : W  (W/A) be a natural projection 

map.  

Then B=   (B/A), and so by [4, prop.27(2), p.1054] B ≤se C. 
But B is an s- closed submodule in W. Thus B=C. 



 

Mathematics |231 
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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

It follows that ( B / A ) = ( C/ A ) and ( B/ A ) is an s-closed submodule in (W/A). 

Proposition 1 . 13 :  
     Let A ≤ B ≤ W such that A is an s -closed submodule of an L-module W. Then B ≤sc W if 

and only if 
 

 
 ≤sc 

 

 
. 

Proof : () See [1, coro.2.7] 

() Suppose 
 

 
 ≤sc 

 

 
  and let B ≤se H ≤ W. Since A ≤sc W and A ≤ B then 

 

 
 ≤se 

 

 
 implies         

B ≤se W by [1, Remarks and Examples 2.2(6)]. That is A ≤sc B by [1, propo.2.8].  

To prove A ≤sc H, suppose that A≤se C for some submodule C of H. As A is an s- closed 

submodule of W, thus A = C . Hence A is an s -closed submodule of H and B ≤se H, that is       
 

 
 ≤se 

 

 
, by [1, Remarks and Examples 2.2(6)]. But 

 

 
 ≤sc 

 

 
, so 

 

 
 = 

 

 
. Then B =H which means 

B ≤sc W. 

Proposition 1 . 14 : 

     Let W be a FMFG L-module, and C≤ W. C is an s-closed submodule in W if and only if C 

= HW for some s-closed ideal H in L. 

Proof: 

() Let C≤ W , then C = HW. To prove H is an s-closed ideal in L . 

Assume H≤se J.  Hence HW≤se JM by (Th. 1.7) , thus C≤se JW so C = JW that is HW =JW. 

Since W is FMFG module so H = J , hence H is an s -closed ideal in L. 

() Similarly. 

2. Ascending (Descending) Chain Conditions on S-Closed Submodules 
     In this section, we study modules with chain conditions on s ˗closed submodules. 

Definition 2 . 1 : An L-module W is said to have the ascending (descending) chain condition , 

briefly A C C (D  C C) on s-closed submodules if every ascending (descending) chain A1  A2  

… ( A1  A2 … ) of s-closed submodules of W is finite. That is there exists k Z+ such   that 

An = Ak for all  nk. 

     Recall that, “a Noetherian module is a module that satisfies the Ascending Chain  

Condition on its submodules. Also, an Artinian module is a module that satisfies the   

Descending  Chain  Condition on its submodules”. [3] 

Remarks 2 . 2 :  

1. Every noetherian (respectively artinian) module satisfies A  C C ( respectively  D C C ) 
on s -closed submodules. 

2. If W satisfies A C C ( respectively D C C) on closed submodules, then W satisfies A C C 
(respectively D C C) on s -closed submodules. 

Proof:  It is clear since every s -closed submodule in W is closed submodule in W. 

     The converse is true if W is hollow by Remark 1.2(6) or uniform module , where “ a   

uniform  module is a nonzero module W which is every non-zero submodule of W is essential 

in W” . [3] 

     Recall that, “an L-module W is called chained if for all submodules C and D of W either   

C ≤ D or D ≤ C”. [7] 

Proposition 2 . 3 : Let W be a chained L -module, and let A be an s-closed submodule of W. 

If W satisfied A C C ( respectively D C C) on s -closed submodules, then A satisfies the A C C 

(respectively D  C  C) on s-closed submodules. 

Proof: Assume W satisfies A C C on s-closed submodules and A1  A2  … be ascending 

chain of s- closed submodules of A. Since A is an s-closed submodule of W and W satisfy 

chained condition, so by [1, prop.2.11, p.345] Ai is an s-closed submodule of W for each i = 

1, 2, … . Hence A1  A2  … be ascending chain of s-closed submodules of W. But W 

satisfies A C C on s- closed submodules , thus kZ+ such that An = Ak for all nk. That is A 

satisfies A C C on s- closed submodules.  

https://en.wikipedia.org/wiki/Module_%28mathematics%29
https://en.wikipedia.org/wiki/Ascending_chain_condition
https://en.wikipedia.org/wiki/Ascending_chain_condition
https://en.wikipedia.org/wiki/Submodule
https://en.wikipedia.org/wiki/Module_%28mathematics%29
https://en.wikipedia.org/wiki/Descending_chain_condition


 

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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

Similarly, if W satisfies D  C C on s- closed submodules, then A satisfies D C C on s- closed 

submodules of A. 

Proposition 2. 4 : Let W = W1⊕W2 be an L-module satisfies A C C (respectively D C C) on s-
closed submodules. Then W1 and W2 satisfy A C C (respectively D  C C) on s- closed 

submodules. 

Proof: Suppose W satisfies A C C (respectively D C C) on s-closed submodule and A1  A2  

…(respectively A1  A2 …)  be ascending (respectively descending) chain of s-closed 

submodules of W1. Thus A1⊕W2, A2⊕ W2, … are s-closed submodules of  W1⊕W2 , by [1, 
prop.2.5]. That is A1⊕W2    A2⊕ W2   … (respectively A1⊕ W2  A2⊕ W2 …) is a 
chain of s-closed submodules of W, but W satisfies A C C (respectively D C C)  on s-closed 

submodules. So there exists kZ+ such that An⊕ W2  = Ak ⊕ W2  for all n  k. So An = Ak for 
all nk. Hence W1 satisfies A C C (respectively D C C)  on s-close submodules. By the same 

way of proof, W2 satisfies A C C (respectively D C C) on s-closed submodules. 

     Recall that , “ a submodule C is fully invariant in W if f(C)  C for all f  EndR(W)”. [3] 

Proposition 2. 5 : Let W = W1⊕W2 be an R-module where W1 and W2 are s-closed 
submodules of W . Then W satisfies A  C C (respectively D C C) on nonzero s- closed 

submodules if and only if W1 and W2 satisfy A C C (respectively D C C)  on nonzero s-closed 

submodules, provided that every s- closed submodule of W is a fully invariant. 

Proof: 

() See proposition 2.4. 

() Suppose W1 and W2 satisfy A C C (respectively D C C) on s-closed submodules, to prove 

W satisfy A C  C (respectively D C C) on s-closed submodules. Let and A1  A2  … 

(respectively A1  A2 …) be ascending (respectively descending) chain of s-closed 

submodules of W.  

Let πi : W  Wi be a projection map for each i = 1, 2. Suppose that Ai = (Ai  ∩ W1) ⊕ (Ai 
∩W2) by [10, Lemma.2.1].  

Note that, Ai  , W1 and W2 are s-closed submodules of W, for each i. Thus by [1, Remarks and 

Examples 2.2 (3)] (Ai  ∩ W1) and (Ai ∩W2) are s-closed submodules of W. Since (Ai  ∩ W1)  

W1  W, so by [1, prop.2.8, p.345]  (Ai  ∩ W1) is an s-closed submodule of W1 and (Ai ∩W2) 

is an s-closed submodule in W2 for each i = 1,2, … . In fact if Ai ∩ Wj = 0 for all i = 1,2, … 

and j = 1,2 then Ai  =  (Ai  ∩ W1) ⊕ (Ai ∩ W2) = 0 which is a contradiction with our 
assumption. That is Ai ∩ Wj are nonzero s-closed submodules in W for each i = 1, 2, … and    

j = 1, 2. So we have the following ascending (respectively descending)  chain of nonzero s-

closed submodules in Wj, (A1  ∩ Wj)  (A2 ∩Wj)  … (respectively A1∩Wj   A2 ∩Wj …) 

for each j = 1, 2. But Wj satisfies A C C (respectively D C C) on s-closed submodules for each    

j = 1, 2. Thus there exists kjZ+ such that   An ∩ Wj  = Akj  ∩ Wj ,for all nkj  and j = 1, 2. Let   

k = max { k1, k2 }, so An = (An  ∩ W1) ⊕ (An ∩W2) = (Ak  ∩ W1) ⊕ (Ak ∩W2) = Ak ,for all 
nk. Hence W satisfies A C C (respectively D  C C). 

Remark 2. 6 : 

     We can generalize proposition 2.5 for finite index I of the direct sum of L-modules. 

Proposition 2. 7 : Let A ≤ B ≤ W such that A is an s-closed submodule of an L- module W. 

W satisfies A C C (respectively D C C) on s-closed submodules if and only if   
 

 
   satisfies A C 

C (respectivelyD C C) on s-closed submodules. 

Proof: () Suppose W satisfied A C C on s- closed submodules, and let  
  
 

   
  
 

   … , be 

ascending chain of s-closed submodules of  
 

 
, then Bi is an s-closed submodule of W by 

(proposition 1.12) . Thus there exists k  Z+ such that Bn = Bk for all nk . Hence 
  
 

 =  
  
 

  for 

all nk .That is 
 

 
  satisfies A C C on s-closed submodules. 



 

Mathematics |233 
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 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

() Suppose 
 

 
  satisfies A C C on s- closed submodules. Let A  A1  A2  … be a chain of 

s-closed submodules of W. Since A  A1 and A  A2 , … and A is an s-closed submodule of 

W, then by [1, coro.2.7, p.345] 
  
 

 is an s-closed submodule of W for each i. Thus we have 
  
 

 

  
  
 

   … is an ascending chain of s-closed submodules of  
 

 
 , hence by our assumption 

 

 
 satisfies A C C on s-closed submodules so there exists k  Z+ such that 

  
 

 =  
  
 

 for all nk. 

That is An = Ak for all nk which means W satisfied A C C on s-closed submodules. 

By the same way we can prove that W satisfies D C C on s-closed submodules if and only if  
 

 
  

satisfies D C C on s-closed submodules. 

Proposition 2. 8 : Let W = W1 ⊕ W2 be an L-module and L = ann(W1) + ann(W2). Then W 
satisfies A C C (respectively D  C C) on s- closed submodules if and only if W1 and W2 satisfy A 

C C (respectively D C C) on s- closed submodules. 

Proof: () see proposition 2.4. 

() Let E1  E2  … be an ascending chain of s- closed submodules of W (Since L = 

ann(W1) + ann(W2), every submodule Ei of W has the form Ni⊕Ki for some Ni ≤ W1 and Ki ≤ 
W2). Hence by [1, prop.2.5] Ni is an s- closed submodule in W1 ,and Ki is an s- closed 

submodule of W2 for all i= 1, 2, … .So N1  N2  … is an ascending chain of s- closed 

submodules of W1 and K1  K2  … is an ascending chain of s- closed submodules of W2. 

Since W1 and W2 satisfy A C C on s- closed submodules, then there exists  t, r  Z+ such that 

Nt =Nt+i and Kr = Kr+i , for each i = 1, 2, … . Take s = max {t, r}, hence Ns⊕Ks  Ns+i⊕Ks+i, 
for each i = 1, 2, … . That is W satisfies A C C on s- closed submodules. 

By the same way we can prove that W satisfies D  C C on s- closed submodules if and only if 

W1 and W2 satisfy D  C C on s- closed submodules. 

Proposition 2. 9 : Let W be an L-module such that the sum of any two s- closed submodules 

of W is again an s- closed submodule. If A is an s- closed submodule of W such that A and  
 

 
 

satisfy A C C (respectively D C C) on s-closed submodules, then W satisfies A C C (respectively 

D C C) on s- closed submodules. 

Proof: Assume B1  B2  … be ascending chain of s -closed submodules of an L-module W, 

then by [1, Remaks and Examples 2.2(3), p.343] Bi ∩ A is an s-closed submodule of W, for 

each i = 1, 2, … , but (Bi∩A)  A, thus Bi∩A is an s-closed submodule of A, for each i = 1, 2, 

… ,  by [1, prop. 2.8, p.345]. 

Also, Bi + A is an s - closed submodule of W (by our assumption), hence 
       

 
 is an s - closed 

submodule of  
 

 
 , for each i = 1, 2, … ,  by proposition 1.12. 

Now consider the two following two ascending chain of s-closed submodules of A and 
 

 
: 

B1 ∩ A  B2 ∩ A  … , and  
       

 
  

       

 
  … , but A and 

 

 
 satisfy A C C on s-closed 

submodules. Therefore , there exists k1, k2  Z+ such that Bn ∩ A = Bk1 ∩ A, for each n  k1, 

and  
       

 
 = 

        

 
 , for each n  k2. By isomorphism theorem 

   

 
  

 

     
 [2, Th. 3.4.3, p. 

56] , so 
       

 
  

  
      

. 

Hence, 
  

      
 = 

   
       

  , which means Bn ∩ A = Bk2 ∩ A , for each n  k2. Let k = max{ k1, k2 

}, thus Bn ∩ A = Bk ∩ A for each n  k and Bn ∩ A = Bk ∩ Bn for each n  k. 

Now, for each n  k , Bn= Bn ∩ (Bn + A) = Bn∩ ( Bk + A) = Bk ∩ ( Bk + A) = Bk . 

Thus, M satisfies A C C on s-closed submodules. 

By a similarly proof W satisfies D C C on s-closed submodules. 



 

Mathematics |234 
https://doi.org/10.30526/30.3.1618  

 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

Proposition 2. 10 : Let W be a FMFG L-module. Then W satisfies A C C (respectively D C C) 

on s- closed submodules if and only if L satisfies A  C C (respectively D C C) on s- closed 

ideals. 

Proof: () Suppose W satisfies A C C (respectively D  C C) on s- closed submodules. To prove 

L satisfies A C C (respectively D C C) on s- closed ideals. Let I1  I2  … (I1  I2  … ) be an 

ascending (respectively descending) chain of s-closed ideals of L. Thus by (proposition 1.14) 

A1 = I1W  A2 = I2W  … (respectively A1 = I1W  A2 = I2W  …) is an ascending 

(respectively descending) chain of s-closed submodules of W. But W satisfies A  C C 

(respectively D C C) on s-closed submodules, so there exists  k  Z+ such that An = Ak for all 

nk, hence InW = IkW for all nk, that is In = Ik for all nk . So L satisfies A C C (respectively 

D C C) on s- closed ideals. 

() Similarly. 

     Recall that, “an L- module W is called a scalar module if every L- endomorphism of W is 

a scalar homomorphism, that is for each 0 ≠ f End(W), there exists 0 ≠ s L such that    

f(a)= sa for all aW”. [11] 

Corollary 2. 11 : Let W be a FMFG L-module. Then W satisfies A C C (respectively D C C) on 

s- closed submodules if and only if End(W) satisfies A  C C (respectively D C C) on s- closed 

ideals. 

Proof: () Since W be a FMFG L-module, then W is a scalar module by [11, Coro.1.1.11], 

End(W)  
 

      
 by [12, Lemma 6.2]. But ann(W) = 0, so End(W)  L. Hence the result 

follows by proposition 2.10. 

() Similarly. 

Future works: 

1. Give an example shows that every noetherian (respectively artinian) module satisfies 
ACC ( respectively  D C C ) on s -closed submodules. 

2. Give an example shows that the converse of (Remark 2.2(2)) is not true in general. 
3. Give an example shows that the converse of (Proposition 2.4) is not true in general. 
 

References 
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International Journal of Scientific and Technical Research,  6, 1. 

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Southeast Asian Bulletin of Mathematics, 35, (1051-1062).  

5. Fleury, P. (1974), “Hollow Modules and local Endomorphism rings”, Pacific J. Math. 53, 

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Mathematics |235 
https://doi.org/10.30526/30.3.1618  

 7102( عام 3( العدد ) 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                           المجلد ) 

Ibn Al-Haitham J. for Pure & Appl. Sci.                                           Vol.03 (3) 2017 

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