Communications on Applied Nonlinear Analysis 

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An Approach to Goldie Extending Modules on the Class of Cyclic 

Submodules 

 

Abdallah Shihadeh 

Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, PO box 330127, Jordan 

abdallaha_ka@hu.edu.jo 

 

Article History: 

Received: 12-04-2024 

Revised: 18-05-2024 

Accepted: 01-06-2024 

Abstract:   In this article, we provide a class of modules that is comparable to 𝐺𝑧-extending 

and 𝐺𝑟-extending modules. We specify what a module M is as 𝐺𝑝-extending if and only if 

for each cyclic submodule A of M, there exists a direct summand D of M such that A∩ 𝐷 

is essential in both A and D. We look into 𝐺𝑝-extending modules and locate this inference 

between the other extending properties. We present some of characterizations of 𝐺𝑝-

extending condition. We show that the direct sum of 𝐺𝑝-extending need not be 𝐺𝑝-

extending and deal with decompositions for 𝐺𝑝-extending concept."   

Keywords: cyclic submodules, P-extending  modules, Goldie extending modules, 𝐺𝑝-

extending. 

1. Introduction 

Throughout this paper, all rings are associative with unitary, R denotes such a ring, and all modules 

are unital right R- modules. In the spirit of [1] , for a module M, think of the following relations on the 

set of submodules of M:" 

𝐴𝛼𝐵 if and only if there exists a submodule C of M such that 𝐴 ≤e C and 𝐵 ≤e C. 

𝐴𝛽𝐵 if and only if 𝐴 ∩ 𝐵 ≤e A and 𝐴 ∩ 𝐵 ≤e B . Recall that β is an equivalence relation. "It is clear 

that a module M is extending (or CS) if and only if for each submodule A of M, there is a direct 

summand D of M such that AαD, (see [1,2]). Further a module M is called Goldie extending module 

(or G-extending) if and only if for each submodule A of M , there is a direct summand D of M such 

that AβD or equivalently, for each closed submodule A in M, there is a direct summand D of M such 

that AβD (see [1]). Obviously, every extending module is G-extending.  

As a generalization of CS-modules is p-extending (see [3,4]). Recall that a module M is called p-

extending if every cyclic submodule of M is essential in a direct summand of M."  

    "In this paper, we study a module condition including the β relation on the set of all cyclic 

submodules of a module. We call a module M is 𝐺𝑝-extending if for every cyclic submodule A of M, 

there is a direct summand D of M such that AβD. A ring R is 𝐺𝑝-extending if 𝑅𝑅 is 𝐺𝑝-extending 

module. It is clear that the class of 𝐺𝑝-extending modules property contains the type of G-extending 

modules. The notion of 𝐺𝑝-extending generalizes both of G-extending, extending and p-extending 

modules." 

     "In section 2, we consider connections between 𝐺𝑝-extending property, p-extending and G-

extending conditions. Moreover, we give sufficient circumstances under which p-extending and 𝐺𝑝-

extending modules are equivalent. 



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Section 3, is devoted to the characterizations of 𝐺𝑝-extending modules. Since the direct sum of 𝐺𝑝-

extending modules need not be 𝐺𝑝-extending, we focus when a direct sum of 𝐺𝑝-extending modules 

is also 𝐺𝑝-extending. Also, we give sufficient conditions under which the direct summand of 𝐺𝑝-

extending is also 𝐺𝑝-extending. These are introduced in section 4. Also, in section 4, we investigate 

𝐺𝑝-extending essential extensions of a module or ring." 

    Following [5], M is called UC-module if every submodule of M has a unique closure in M.  

2.  Preliminary results. 

    "The 𝐺𝑝-extending notion is based on two tools, namely an equivalence relation on cyclic 

submodules of a module M.  Let us begin by mentioning basic facts about them. First recall the 

following relations on the set of submodules of M (see [1]). 

(i) 𝐴𝛼𝐵 if and only if there exists a submodule C of M such that 𝐴 ≤e C and 𝐵 ≤e C. 

(ii) 𝐴𝛽𝐵 if and only if 𝐴 ∩ 𝐵 ≤e A and 𝐴 ∩ 𝐵 ≤e B. 

Observe that α is reflexive and symmetric, but it may not be transitive. However, β is an equivalence 

relation. Note that for submodules of module M. if AαB, then AβB.  

Proposition 2.1: A module M is p-extending if and only if for each cyclic submodule A of M, there is 

a direct summand D of M such that A𝛼 D. 

Proof:  The proof is routine." 

"Motivated by proposition 2.1 and Akalan, Birkenmeier, Tercan's use of the β equivalence relation in 

[1]. As a generalization of Goldie extending modules, we introduce a class of modules which is 

analogous to that of 𝐺𝑧-extending and 𝐺𝑟-extending modules which are introduced in [6] and [7] 

respectively." 

Definition 2.2: We call a module M is 𝐺𝑝-extending module if for each cyclic submodule A of M, there 

is a direct summand D of M such that 𝐴𝛽𝐷. 

   "Note that M is G-extending if and only if for each submodule A of M there is a direct summand D 

of M such that 𝐴𝛽𝐷. It is clear that the class of 𝐺𝑝-extending contains both of the classes of G-

extending and p-extending modules. 

   Now, we locate the 𝐺𝑝-extending condition with respect to several known generalizations of the 

extending property." 

Proposition 2.3: Make M a module. Let's think about the aforementioned circumstances. 

(i) M is CS. 

(ii) M is G-extending. 

(iii)  M is 𝐺𝑝-extending. 

(iv) M is p-extending. 

 



Communications on Applied Nonlinear Analysis 

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"Then (i) ⟹(ii) ⟹(iii) and (i) ⟹ (iv)⟹(iii). In general, the converse implications do not hold." 

Proof: (i) ⟹(ii) ⟹(iii) and (i) ⟹ (iv)⟹(iii) are clear. 

(ii)⇏(i)"Let M be the 𝕫-module 𝕫𝑝⨁ℚ, where p is any prime integer. Then 𝑀ℤ is G-extending by [1, 

corollary (3.3)]. However 𝑀ℤ is not extending [8 , Example 10]." 

(iii)⇏(ii) Let 𝑀2(𝑅)be the ring as in [9, Example 13.8]. Then 𝑀2(𝑅) is a von Neumann regular ring 

which is not a Baer ring. Hence it is neither right nor left CS, by [10, example 2.7] , however it's well 

acknowledged that each von Neumann regular ring is nonsingular, therefore 𝑀2(𝑅) is not is G-

extending, see [1, Proposition 1.8]. Also, this is an example to show that (iv)⇏(i).   

 (iii)⇏(iv) Let M be the 𝕫-module 𝕫2⨁𝕫8. Then 𝑀ℤ is 𝐺𝑝-extending but not P-extending, see [1, 

Corollary 3.3]. 

   The condition under which 𝐺𝑝-extending and p-extending modules are equivalent is stated in the 

following proposal. 

Proposition 2.4:" Let M be a module. 

(i) If M is a UC- module. Then M is 𝐺𝑝-extending if and only if M is P-extending. 

(ii) If M is a nonsingular module. Then M is 𝐺𝑝-extending if and only if M is P-extending. 

(iii) If M is an indecomposable module. Then M is 𝐺𝑝-extending if and only if M is P-

extending." 

Proof:  

(i) "Assume that M is 𝐺𝑝-extending and let A be a cyclic submodule of M, then 

there exists a direct D of M such that 𝐴𝛽𝐷. One can easily show that (𝐴 ∩ 𝐷)𝛼𝐴 

and (𝐴 ∩ 𝐷)𝛼𝐷. But M is UC module, therefore α is transitive, hence 𝐴𝛼𝐷. 

Thus M is P-extending. The converse is clear. 

(ii)  Let M be a 𝐺𝑝-extending and let A be a cyclic submodule of M, then there is a 

direct D of M such that 𝐴𝛽𝐷. It is sufficient to show that 𝐴 ≤ 𝐷. Since 
𝐴+𝐷

𝐷
≅

𝐴

𝐴∩𝐷
 is singular and 

𝐴+𝐷

𝐷
≤

𝑀

𝐷
≅ 𝐷′ is nonsingular, hence A+D = D which implies 

that 𝐴 ≤ 𝐷. The converse is obvious.  

(iii)  Let M be a 𝐺𝑝-extending and let A be a cyclic submodule of M, then there is a 

direct D of M such that 𝐴𝛽𝐷. Since M is indecomposable, then D=M. Thus M 

is P-extending module. The converse is clear." 

Corollary 2.5: "Let M be an indecomposable module. Then the following statements are equivalent: 

(i) M is uniform. 

(ii) M is CS. 

(iii) M is G-extending. 

(iv) M is P-extending. 



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(v) M is 𝐺𝑝-extending." 

Example 2.6:  

"Let F be a field and V be a vector space over F with dim (𝐹𝑉)=2. Let R be the trivial extension of F 

with V, i.e,  

𝑅 = [
𝐹 𝑉
0 𝐹

] = {[
𝑓 𝑣
0 𝑓

] : 𝑓 ∈ 𝐹, 𝑣 ∈ 𝑉}. Since 𝑅𝑅 is indecomposable which is not CS, then  𝑅𝑅 is not 

𝐺𝑝-extending." 

3. Characterizations of 𝐺𝑝-extending. 

    "In this section, we give equivalent conditions to 𝐺𝑝-extending property.  

We start by the following theorem. 

Theorem 3.1: An R- module M is 𝐺𝑝-extending if and only if for each cyclic submodule A of M, there 

is a direct summand D of M such that 𝐴𝛽𝐷 and D' is a complement of A, where 𝑀 = 𝐷⨁𝐷′. 

Proof: Suppose that M is 𝐺𝑝-extending , let A be a cyclic submodule of M, there is a direct summand 

D of M such that 𝐴𝛽𝐷. Let 𝑀 = 𝐷⨁𝐷′, for some submodule D' of M. Since 𝐴 ∩ 𝐷 ≤e A , then 𝐴 ∩

𝐷′ = 0. Now, let B be a submodule of M such that 𝐷′ ≤ 𝐵 and 𝐴 ∩ 𝐵 = 0. Since 𝐴 ∩ 𝐷 ≤e D, then 

𝐵 ∩ 𝐷 = 0. But D' is a complement of D, therefore B=D'. Thus, D' is a complement of A. The converse 

is clear."  

      "The next result gives another characterization to 𝐺𝑝-extending modules. 

Proposition 3.2: Let M be an R- module, the following conditions are equivalent: 

(i) M is 𝐺𝑝-extending. 

(ii) For all cyclic submodule A of M , there exists a submodule X of M and a direct summand D of 

M such that 𝑋 ≤e A and 𝑋 ≤e D. 

(iii)  For every cyclic submodule A of M there exists a complement B of A and a complement C of 

B such that AβC and each homomorphism 𝑓: 𝐶⨁𝐵 → 𝑀 extends to a homomorphism 𝑔: 𝑀 →

𝑀. 

Proof: (i) ⟹(ii) Assume that M is 𝐺𝑝-extending and let A be a cyclic submodule of M, there is a direct 

summand D of M such that 𝐴𝛽𝐷, hence 𝐴 ∩ 𝐷 ≤e A and 𝐴 ∩ 𝐷 ≤e D. Take X = 𝐴 ∩ 𝐷, we get the 

result. 

(ii)⟹(iii) Let A be a cyclic submodule of M. By (ii), there exists a submodule X of M and a direct 

summand D of M such that 𝑀 = 𝐷⨁𝐷′, 𝑋 ≤e A and 𝑋 ≤e D. Take D = C and D' = B. 

(iii)⟹(i) Let A be a cyclic submodule of M. From (iii), there exists a complement B of A and a 

complement C of B such that AβC and every homomorphism 𝑓: 𝐶⨁𝐵 → 𝑀 extends to a 

homomorphism 𝑔: 𝑀 → 𝑀and by [11, Lemma 3.97], D is a direct summand of M, hence M is 𝐺𝑝-

extending." 

 



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Theorem 3.3 : A module M is 𝐺𝑝-extending if and  only if for every direct  summand A of the injective  

hull E(M) of M with 𝐴 ∩ 𝑀 is cyclic submodule  of M, there is a direct  summand D of M such  that 

(𝐴 ∩ 𝑀)𝛽𝐷. 

Proof: "Let A be a cyclic submodule of M and let B be a complement of A, then 𝐴⨁𝐵 ≤e M. Since 

𝑀 ≤e E(M), then 𝐴⨁𝐵 ≤e E(M) implies 𝐸(𝑀) = 𝐸(𝐴)⨁𝐸(𝐵). It can be seen that 𝐸(𝐴) ∩ 𝑀 is cyclic 

submodule in M. By our assumption, there is a direct summand D of M such that (𝐸(𝐴) ∩ 𝑀)𝛽𝐷. But 

we have(𝐴 ∩ 𝑀)𝛽(𝐸(𝐴) ∩ 𝑀), hence 𝐴𝛽𝐷. The converse implication is clear." 

Theorem 3.4:" Suppose M is an R-module. The assertions that follow are identical. 

(i) M is 𝐺𝑝-extending module. 

(ii) A decomposition exists for each cyclic submodule A of the module M. M = DD', such that 

(D'+A) βM. 

(ii) Fore  very cyclic  submodule A of M, there is  a decomposition A

M

= A

L

 A

K

such  that L is a 

direct summand of M and KβM." 

Proof: (i)(ii)"Let M be a 𝐺𝑝-extending and let A be a cyclic submodule of M, there exists direct 

summand D of M such that A βD, then M = DD', D' ≤ M. Since {A, D'} is an independent family, 

then (A+D') βM , see [12, Proposition 1.4 ]." 

(ii)(iii) "Let A be a cyclic submodule of M. By (ii), there is a decomposition M = DD', such that 

(D'+A) βM. Claim that A

M

= A

AD +

 A

AD +'

. Since M = DD', then A

M

= A

DD '+

= A

D

+ A

AD +'

 and 

A

AD +

 A

AD +'

= A

ADD )'( +

= A

DDA )'( +

=A, hence A

M

= A

AD +

 A

AD +'

. Take K = D'+A and 

L = D+A, so we get the result." 

(iii)(i)"To show that M is 𝐺𝑝-extending, let A be a cyclic submodule of M. By (iii), there is a 

decomposition A

M

= A

L

 A

K

such that L is a direct summand of M and KβM. It is enough to show 

that A βL. Let i :L→ M be the injection map. Since KβM, then i -1 (K) βi -1 (M), that is (LK)βD. One 

can easily show that LK = A, so M is 𝐺𝑝-extending  module." 

"Proposition 3.5: Let M be an R-module. Then M is 𝐺𝑃-extending module if and only if for every 

cyclic submodule A of M, there exists an idempotent f End (M) such that A βf (M)." 

4. Decompositions. 

"There are nonsingular modules 𝑀 = 𝑀1⨁𝑀2 in which 𝑀1 and 𝑀2 are P-extending, but M is not P-

extending (e.g, Let R = 𝕫[𝑥] be a polynomial ring of integers and let M = 𝕫[𝑥]⨁𝕫[𝑥]). Note that 𝕫[𝑥] 

is 𝐺 −extending, by [1] and hence 𝐺𝑝-extending but M is not P-extending which is nonsingular, thus 

by proposition 2.4 M is not 𝐺𝑝-extending. Next, we give various conditions under which the direct 

sum of 𝐺𝑝-extending is 𝐺𝑝-extending." 



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Proposition 4.1: Let M = M1M2 be a distributive module if M1 and M2 are 𝐺𝑝-extending modules, 

then M is 𝐺𝑝-extending. 

Proof: "Let A be a cyclic submodule of M. Since M is distributive, then A = A∩M = A∩(M1M2) = 

(A∩M1) (A∩M2). Since A is cyclic in M, then A∩M1 and A∩M2 are cyclic in M1 and M2 respectively. 

But M1 and M2 are 𝐺𝑝-extending modules, therefore there are direct summand 𝐷1 of 𝑀1 and 𝐷2 of 𝑀2 

such that (𝐴⋂𝐷1)𝛽𝐷1 and (𝐴⋂𝐷2)𝛽𝐷2 hence A β(𝐷1⨁𝐷2), by [12 , Proposition 1.4]. Thus, M is 𝐺𝑝-

extending module."  

   The following statements are also easily proved by using a similar argument. 

Proposition 4.2 : "Let M = M1M2 be a duo module if M1 and M2 are 𝐺𝑝-extending modules, then M 

is 𝐺𝑝-extending." 

Proposition 4.3: Let M1 and M2 be 𝐺𝑝-extending modules such that annM1+annM2 = R, then M1M2 

is 𝐺𝑝-extending module. 

Proposition 4.4: "Let M = M1M2 be an R- module with M1 being 𝐺𝑝-extending and M2 is 

semisimple. Suppose that for any cyclic submodule A of M, A∩M1 is a direct summand of A, then M 

is 𝐺𝑝-extending. 

Proof: Let A be a cyclic submodule of M, then it is easy to see that A+M1 = M1 [(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. One can easily show that A∩M1 is cyclic in M1. But M1 is 𝐺𝑝-extending, then there is a direct 

summand D of M1 such that (A∩M1) β D is hence A = ((A∩M1)A')β(M1+A). Thus, M is 𝐺𝑝-

extending." 

Proposition 4.5: "Let M = M1M2 such that M1 is 𝐺𝑝-extending and M2 is injective module. Then M 

is 𝐺𝑝-extending if and only if for every cyclic submodule A of M such that A∩M2≠0 there is a direct 

summand D of M such that 𝐴𝛽𝐷. 

Proof: Suppose that for every cyclic submodule A of M such that A∩M2≠0 there exists direct summand 

D of M such that 𝐴𝛽𝐷. Let A be a cyclic submodule of M such that A∩M2 = 0. By [2], there is a 

submodule M' of M containing A such that M = M'M2. Since M'≅
𝑀

𝑀2
≅M1 is 𝐺𝑝-extending and A is 

cyclic submodule of M', then there is a direct summand K of M' such that AβK. Thus, M is 𝐺𝑝-

extending. The converse is obvious." 

   We now list several circumstances in which a direct summand of a module that extends 𝐺𝑝-extending 

is 𝐺𝑝-extending. 

Proposition 4.6:"Let A be a direct summand of a 𝐺𝑝-extending module M , if the intersection of A 

with any direct summand of M is a direct summand of A, then A is 𝐺𝑝-extending module." 

Proof: "Let X be a cyclic in A, then X is cyclic in M. But M is 𝐺𝑝-extending, therefore there exists a 

direct summand D of M such that XβD. It can be seen that Xβ (AD). By our assumption AD is a 

direct summand of A. Thus, A is 𝐺𝑝-extending." 



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Proposition 4.7:"Let A be a cyclic submodule of a 𝐺𝑝-extending module M ." 

(i) If for each 𝑒2 = 𝑒 ∈ 𝐸𝑛𝑑(𝑀𝑅), there exists 𝑓2 = 𝑓 ∈ 𝐸𝑛𝑑(𝐴𝑅) such that 𝐴 ∩ 𝑒𝑀 ≤𝑒 𝑓𝐴, 

then A is 𝐺𝑝-extending. 

(ii) If for each 𝑒2 = 𝑒 ∈ 𝐸𝑛𝑑(𝑀𝑅), there exists 𝑓2 = 𝑓 ∈ 𝐸𝑛𝑑(𝐴𝑅) such that 𝑒𝑀𝛽𝑓𝑀 and 

𝑓𝐴 ⊆ 𝐴, then A is 𝐺𝑝-extending. 

Proof:  

(i) "Let Y be a cyclic submodule of A. Hence Y is a cyclic submodule of M. By proposition 

3.2, there is X≤e Y and 𝑒2 = 𝑒 ∈ 𝐸𝑛𝑑(𝑀𝑅) such that X≤e eM. Then X≤e eM∩ 𝐴 ≤e f A, for 

some 𝑓2 = 𝑓 ∈ 𝐸𝑛𝑑(𝑀𝑅).Thus, A is 𝐺𝑝-extending." 

(ii) "Let Y be a cyclic submodule of A, then Y is cyclic in M. Then there exists 𝑒2 = 𝑒 ∈

𝐸𝑛𝑑(𝑀𝑅) such that YβeM. Hence YβfM . Since fA⊆ 𝐴 , A is 𝐺𝑝-extending." 

Proposition 4.8:"Let K be a projection invariant cyclic submodule of M. If M is 𝐺𝑝-extending, then 

there exists 𝑀1 ≤ 𝑀 such that 𝑀 = 𝑀1⨁𝐾 and K is 𝐺𝑝-extending. 

Proof: There exists 𝑒2 = 𝑒 ∈ 𝐸𝑛𝑑(𝑀𝑅) such that 𝐾𝛽𝑒𝑀. But 𝐾 = 𝑒𝐾⨁(1 − 𝑒)𝐾, 𝑒𝐾 = 𝐾 ∩ 𝑒𝑀, and 

(1 − 𝑒)𝐾 = 𝐾 ∩ (1 − 𝑒)𝑀 because K is projection invariant, then 𝑒𝐾 ≤𝑒 𝑒𝑀 and 𝑒𝐾 ≤𝑒 𝐾. Hence 

𝐾 ∩ (1 − 𝑒)𝑀 = 0. So 𝐾 = 𝑒𝐾 ≤𝑒 𝑒𝑀. Since K is cyclic in M, then K=eM. Let 𝑀1 = (1 − 𝑒)𝑀. 

Therefore 𝑀 = 𝑀1⨁𝐾. Observe that, by Proposition 4.7 (ii), K is 𝐺𝑝-extending." 

Theorem 4.9: Let M be a 𝐺𝑝-extending module. If M has SIP or satisfies the 𝐶3 condition, then any 

cyclic direct summand of M is 𝐺𝑝-extending. 

Proof: "Let 𝑀 = 𝑁⨁𝑁′ for some submodules N, N' of M where N is cyclic in M. Using Proposition 

4.8(i) , where N is taken to be cyclic in M and applying the SIP gives that N is a 𝐺𝑝-extending. 

 Now assume that M satisfies the 𝐶3 condition. Let 𝜋: 𝑀 → 𝑁 be the canonical projection. Let K be 

any cyclic submodule of N, then K is cyclic in M. By hypothesis, there exists a direct summand L of 

M such that 𝐾 ∩ 𝐿 ≤𝑒 𝐾 and ∩ 𝐿 ≤𝑒 𝐿 . Since M satisfies 𝐶3 condition, 𝑁′⨁𝐿 is a direct summand of 

M. It can be seen that 𝑁′⨁𝐿 = 𝑁′⨁𝜋(𝐿)(see [11, Lemma 2.71]). Hence 𝜋(𝐿) is a direct summand of 

N. For any 0 ≠ 𝑦 ∈ 𝜋(𝐿), 𝑦 = 𝜋(𝑥) for some 0 ≠ 𝑥 ∈ 𝐿. There exists an 𝑟 ∈ 𝑅 such that 0 ≠ 𝑥𝑟 ∈

𝐾 ∩ 𝐿. So 𝑥𝑟 = 𝑘 = 𝑥1, where 𝑘 ∈ 𝐾 and 𝑥1 ∈ 𝐿. Now 0 ≠ 𝑥𝑟 = 𝜋(𝑥)𝑟 = 𝑘 = 𝜋(𝑥1) ∈ 𝐾 ∩ 𝜋(𝐿). It 

follows that 𝐾 ∩ 𝜋(𝐿) ≤𝑒 𝜋(𝐿). It is clear that 𝜋(𝐿) = 𝑁 ∩ (𝑁′⨁𝜋(𝐿)) = 𝑁 ∩ (𝑁′⨁𝐿). Hence 𝐾 ∩

𝜋(𝐿) = 𝐾 ∩ (𝑁′⨁𝐿) ≤𝑒 𝐾. Thus, N is 𝐺𝑝-extending."  

  "Next, we investigate 𝐺𝑝-extending essential extensions of a module or ring. Let us begin with the 

following useful result which provides relative injectivity or certain direct summands of a Goldie 

extending module (or nonsingular 𝐺𝑝-extending module)." 

    "Let N, M be modules. N is said to be M-ejective if, for each 𝐾 ≤ 𝑀 and each homomorphism 

𝑓: 𝐾 ⟶ 𝑁, there exists a homomorphism 𝑔: 𝑀 → 𝑁 and 𝑋 ≤𝑒 𝐾 such that g(x) = f (x), for all 𝑥 ∈ 𝑋, 

see [1]." 

Proposition 4.10 : "Let R be any ring , 𝑀1 a semisimple right R- module, and 𝑀2 a right R- module 

with zero socle such that 𝑀 = 𝑀1⨁𝑀2 is a Goldie extending UC- module. Then 𝑀1 is 𝑀2 ejective." 



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Proof: "Obviously, 𝑀1 = 𝑆𝑜𝑐(𝑀). Let N be any submodule of 𝑀2, and let 𝜑: 𝑁 → 𝑀1 be a 

homomorphism. Let 𝐿 = {𝑥 − 𝜑(𝑥): 𝑥 ∈ 𝑁}. Then L is a submodule of M and 𝐿 ∩ 𝑀1=0. There exists 

submodules K, K' of M such that 𝑀 = 𝐾⨁𝐾′, 𝐾 ∩ 𝐿 ≤𝑒 𝐿 and 𝐾 ∩ 𝐿 ≤𝑒 𝐾. It is clear that K is a 

closure of 𝐾 ∩ 𝐿 in M. By assumption, 𝐿 ≤ 𝐾. Since 𝐾 ∩ 𝐿 ∩ 𝑀1 = 𝐿 ∩ 𝑀1 = 0, 𝐾 ∩ 𝐿 ∩ 𝑆𝑜𝑐(𝑀) =

𝑆𝑜𝑐(𝐿) = 0. It follows that 𝑆𝑜𝑐(𝐾) = 𝐾 ∩ 𝑀1 = 0. Hence 𝑀1 = 𝑆𝑜𝑐(𝑀) ⊆ 𝐾′. Thus, 𝐾′ =

𝑀1⨁(𝐾′ ∩ 𝑀2) and 𝑀 = 𝐾⨁𝑀1(𝐾′⋂𝑀2). Let 𝜋: 𝑀 → 𝑀1 denote the canonical projection with 

kernel 𝐾⨁(𝐾′⋂𝑀2). Let 𝜃 be the restriction of 𝜋 to 𝑀2. Then 𝜃: 𝑀2 → 𝑀1. Let x be any element of 

N. Since 𝑥(𝑥 − 𝜑(𝑥)) + 𝜑(𝑥), 𝜃(𝑥) = 𝜑(𝑥). It follows that 𝑀1 is 𝑀2- injective." 

Corollary 4.11: (i)"Let 𝑀 = ⨁𝑖=1
𝑛 𝑀𝑖, where each 𝑀𝑖 is uniform. If 𝐸(𝑀𝑖) ≇ 𝐸(𝑀𝑗) for all 𝑖 ≠ 𝑗, then 

M is 𝐺𝑝-extending. 

(ii) Let S be a simple module and 𝑀1, 𝑀2 ≤ 𝐸(𝑆). If there exists a homomorphism ℎ: 𝑀2 → 𝑆 such 

that ℎ(𝑆) ≠ 0, then 𝑀 = 𝑀1⨁𝑀2 is 𝐺𝑝-extending. 

Proof: (i) From [1, Corollary 4.11] , M is Goldie extending. Thus Proposition 2.3 gives that M is 𝐺𝑝-

extending.  

(ii) By [1, Corollary 4.14], 𝑀1 is 𝑀2- ejective and so it is G-extending. Now, by proposition 2.3 M is 

𝐺𝑝-extending." 

Example 4.12: (i) "Let M be the 𝕫-module (𝕫/𝕫𝑝)⨁ℚ and let T be the polynomial ring 𝕫[𝑥]. Then 𝑀𝕫 

is included in corollary 4.11(i). On the other hand , it is well known that 𝑇2 is not 𝐺𝑝-extendingT- 

module. Hence, we obtain that the condition 𝐸(𝑀𝑖) ≇ 𝐸(𝑀𝑗) for all 𝑖 ≠ 𝑗, is not superfluous in 

corollary 4.11(i). 

(ii) Let K be a field and R=K[x, y] , the commutative local Frobenious K-algebra (see[1, Example 4.15]) 

defined by the relations 𝑥𝑦 = 𝑥2 − 𝑦2 = 0. Then 𝑅𝑅 is a uniform injective module with simple 

submodule 𝐾𝑥2. Let 𝑀2 = 𝑥𝑅 = {𝑘1𝑥 + 𝑘.2 𝑥2: 𝑘𝑖 ∈ 𝐾},  and let h be the R-homomorphism, ℎ: 𝑥𝑅 →

𝐾𝑥2, defined by ℎ(𝑘1𝑥 + 𝑘2𝑥2) = 𝑘2𝑥2. Then ℎ(𝐾𝑥2) ≠ 0. Thus, by Corollary 4.11(ii), 𝑀 =

𝑀1⨁𝑥𝑅 is 𝐺𝑝-extending for any 𝑀1 ≤ 𝑅𝑅." 

   " Next example exhibits that 𝐺𝑝-extending property is not closed under essential extensions of a 

module." 

Example 4.13: "Let F be any field and 𝑅 = [
𝐹 𝐹 𝐹
0 𝐹 0
0 0 𝐹

]. Then 𝑆𝑜𝑐(𝑅𝑅) ≤𝑒 𝑅𝑅. Obviously 𝑆𝑜𝑐(𝑅) is 

a 𝐺𝑝-extending right R- module. However, it is well known that  𝑅𝑅 is not 𝐺𝑝-extending (see [13, 

Theorem 3.4])." 

      " In contrast to essential extensions of a module which satisfies 𝐺𝑝-extending condition, we have 

the following overring of a ring R if Sis an overring of R such that 𝑅𝑅 essential in 𝑆𝑅." 

Theorem 4.14:"Let S be a right essential overring of R (i.e., 𝑅𝑅 ≤𝑒 𝑆𝑅). If 𝑅𝑅 is 𝐺𝑝-extending, then 𝑆𝑅 

and 𝑆𝑆 are 𝐺𝑝-extending. 

Proof: Let 𝑌𝑅 be any cyclic submodule of 𝑆𝑅. That much is clear to see.𝑋 = 𝑌 ∩ 𝑅 is cyclic submodule 

of 𝑅𝑅 . By Proposition 3.2, there exists 𝐾𝑅 ≤ 𝑅𝑅 and 𝑒2 = 𝑒 ∈ 𝑅 such that 𝐾𝑅 ≤𝑒 𝑋𝑅 and 𝐾𝑅 ≤𝑒 𝑒𝑅𝑅. 



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Notice that𝐾𝑅 ≤𝑒 𝑌𝑅 . Now, let us show that 𝐾𝑅 ≤𝑒 𝑒𝑆𝑅. Let 0 ≠ 𝑒𝑠 ∈ 𝑒𝑆. There exists 𝑟1 ∈ 𝑅 such 

that 0 ≠ 𝑒𝑠𝑟1 ∈ 𝑅. Hence 0 ≠ 𝑒𝑠𝑟1 ∈ 𝑒𝑅, so there exists 𝑟2 ∈ 𝑅 such that 0 ≠ 𝑒𝑠𝑟1𝑟2 ∈ 𝐾. Thus 

𝐾𝑅 ≤𝑒 𝑒𝑆𝑅 . By Proposition 3.2, 𝑆𝑅 is 𝐺𝑝-extending. A similar demonstration illustrates that 𝐾𝑆𝑆 ≤𝑒 𝑌𝑆 

and 𝐾𝑆𝑆 ≤𝑒 𝑒𝑆𝑆. Therefore 𝑆𝑆 is 𝐺𝑝-extending. 

Corollary 4.15: Let 𝑇 = 𝑇𝑚(𝑅) and 𝑀 = 𝑀𝑚(𝑅). If 𝑇𝑇 is 𝐺𝑝-extending, then 𝑀𝑇 and 𝑀𝑀 are 𝐺𝑝-

extending. 

Proof: This outcome is a result of Theorem 4.14 and the reality𝑀𝑇 is a rational extension of 𝑇𝑇." 

   "It is not known so far whether direct summands of Goldie extending module enjoy with the property. 

Like the former case the authors desire to obtain whether the 𝐺𝑝-extending property is inherited by its 

direct summands or not? 

Acknowledgments 

The authors would like to express their thanks to the referee for her/his careful reading and useful 

suggestions on this paper." 

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