IHJPAS. 36(1)2023 

300 
 This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

Almost and Strongly Almost Approximately Nearly Quasi Compactly Packed 

Modules 

 

 

 

 
 

 

 
 

 

 

Abstract 

          In this paper we present the almost approximately nearly quasi compactly packed 

(submodules) modules as an application of almost approximately nearly quasiprime submodule. 

We give some examples, remarks, and properties of this concept. Also, as the strong form of this 

concept, we introduce the strongly, almost approximately nearly quasi compactly packed 

(submodules) modules. Moreover, we present the definitions of almost approximately nearly 

quasiprime radical submodules and almost approximately nearly quasiprime radical submodules 

and give some basic properties of these concepts that will be needed in section four of this research. 

We study these two concepts extensively. 

Keywords: Alappnq-prime submodules, Alappnq compactly packed, Strongly Alappnq 

compactly packed, Alappnq-prime radical of submodule, Alappnq-prime radical submodule. 

 

1. Introduction 

 The concept of almost approximately nearly quasiprime was recently introduced by [1] as a 

generation of “quasiprime, nearly quasiprime and approximately quasiprime” submodules see [2-

4]. The submodule 𝐹 of 𝑄 is called almost approximately nearly quasiprime (simply Alappnq-
prime) submodule, if for any 𝑟𝑠𝑞 ∈ 𝐹, for 𝑟,𝑠 ∈ 𝑅, 𝑞 ∈ 𝑄, implying that either 𝑟𝑞 ∈ 𝐹 +
(𝑠𝑜𝑐(𝑄)+𝐽(𝑄)) or 𝑠𝑞 ∈ 𝐹 +(𝑠𝑜𝑐(𝑄)+𝐽(𝑄)). 
As an application of Alappnq-prime submodule, we introduce the concepts of [almost 

approximately nearly quasi compactly packed (submodules) modules, strongly almost 

approximately nearly quasi compactly packed (submodules) modules] and study some basic 

properties of these concepts. This paper consists of three sections. Section one covers some basic 

concepts, recalls some remarks and propositions needed in the sequel. Section two introduces and 

studies the concept of almost approximately nearly quasi compactly packed (submodules) modules 

and gives some basic properties. Section three, devoted to introducing the concept of strongly 

almost approximately nearly quasi compactly packed (submodules) modules. Also, we introduce 

the concepts of almost approximately nearly quasiprime radical submodules and almost 

approximately nearly quasiprime radical submodules and study this concept in detail. Finally, we 

doi.org/10.30526/36.1.3013 

Article history: Received 14 September 2022, Accepted 6 November 2022, Published in January 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Ali Sh. Ajeel 

Department of Mathematics , College of 
Computer Sciences and Mathematics,Tikrit 

University , Iraq. 

Ali.shebl@st.tu.edu.iq 
 

Haibat K. Mohammadali 

Department of Mathematics , College of 
Computer Sciences and Mathematics,Tikrit 

University , Iraq. 
H.mohammadali@tu.edu.iq 

 

https://creativecommons.org/licenses/by/4.0/
mailto:Ali.shebl@st.tu.edu.iq
mailto:H.mohammadali@tu.edu.iq


IHJPAS. 36(1)2023 
 

301 
 

remark that all rings in this paper are commutative with identity and all modules are unitary left 

𝑅-module. 
2.  Preliminaries 

This section includes some well-known definitions, remarks, and propositions needed in our 

study of the next sections. 

Remark 2.1 [5] 

In a finitely generated 𝑅-module, every proper submodule contend in maximal submodule. 

Definition 2.2 [6] 

An 𝑅-module 𝑄 is multiplication if every submodule 𝐹 of 𝑄 is of the form 𝐹 = 𝐼𝑄 for some 
ideal 𝐼 of 𝑅. 

Proposition 2.3 [7] 

Let 𝑄 be a non-zero multiplication 𝑅-module, then every proper submodule of 𝑄 contend in a 
maximal submodule. 

Definition 2.4 [8] 

A submodule 𝐹 of an 𝑅-module 𝑄 is called small if 𝐹 +𝐾 = 𝑄 implies that 𝐾 = 𝑄 for any 
proper submodule 𝐾 of 𝑄. 

Proposition 2.5 [1] 

 Let 𝑓:𝑄 → 𝑄′ be an 𝑅-epimorphism, and ker𝑓 is small submodule for 𝑄. If 𝐹 is an Alappnq-
prime submodule for 𝑄′ then 𝑓−1(𝐹) is Alappnq-prime submodule for 𝑄. 

Proposition 2.6 [1] 

Let 𝑓:𝑄 → 𝑄′ be an 𝑅-epimorphism, and ker𝑓 is small submodule for 𝑄. If 𝐹 be an Alappnq-
prime submodule for 𝑄 with 𝐾𝑒𝑟 𝑓 ⊆ 𝐹 then 𝑓(𝐹) is Alappnq-prime submodule for 𝑄′. 

Definition 2.7 [9] 

A subset 𝑆 of a ring 𝑅 is called multiplicatively closed if 1 ∈ 𝑆 and 𝑎𝑏 ∈ 𝑆 for every 𝑎,𝑏 ∈ 𝑆. 
Let 𝑇 be the set of all order pairs (𝑞,𝑠) where 𝑞 ∈ 𝑄 and 𝑠 ∈ 𝑆. The relation on 𝑇 is defined by 
(𝑞,𝑠)~(𝑞′,𝑠′) if there exists 𝑡 ∈ 𝑆 such that 𝑡(𝑠𝑞′ −𝑠′𝑞) = 0 is an equivalence relation. We 

denote the equivalence classes of (𝑞,𝑠) by 
𝑞

𝑠
. Let 𝑄𝑆 denote the set of all equivalence classes 𝑇 

with respect to this relation. 𝑄𝑆 is an 𝑅-module. 

Definition 2.8 [10] 

An 𝑅-module 𝑄 is 𝑍-regular if for each 𝑞 ∈ 𝑄 there exists 𝑓 ∈ 𝑄′ = 𝐻𝑜𝑚𝑅(𝑄,𝑅) such that 
𝑞 = 𝑓(𝑞)𝑞. 

Proposition 2.9 [5] 

Let 𝑄 be an 𝑅-module then the following statements are equivalent: 

1. Every proper submodule of 𝑄 is a semi prime. 

2. Every proper submodule of 𝑄 is the intersection of prime submodule of 𝑄. 

Proposition 2.10 [5] 

Let 𝑄 be a non-zero 𝑍-regular 𝑅-module, then every proper submodule of 𝑄 is a semi prime. 

From Propositions 2.9 and 2.10, we get the following corollary. 

 

 



IHJPAS. 36(1)2023 
 

302 
 

Corollary 2.11 

Let 𝑄 be 𝑍-regular 𝑅-module, then every proper submodule of 𝑄 is the intersection of a prime 
submodule of 𝑄. 

Remark 2.12 [1] 

Every prime submodule 𝐹 of an 𝑅-module 𝑄 is an Alappnq-prime submodule of 𝑄. 

Definition 2.13 [11] 

An 𝑅-module 𝑄 is faithful if 𝑎𝑛𝑛𝑅(𝑄) = (0). 

Proposition 2.14 [13] 

A proper submodule 𝐹 of faithful multiplication 𝑅-module 𝑄 is an Alappnq-prime submodule 
of 𝑄 if and only if [𝐹:𝑅 𝑄] is an Alappnq-prime ideal of 𝑅. 

Definition 2.15 [9] 

“An 𝑅-module 𝑄 is called Bezout module if every finitely generated submodule of 𝑄 is cyclic”. 

3.  Almost Approximately Nearly Quasi Compactly Packed Modules  

Before we introduce the concept of almost approximately nearly quasi compactly packed 

modules, and study some properties, we need to define the concept of almost approximately nearly 

quasi compactly packed submodules. 

Definition 3.1 

A proper submodule 𝐹 of an 𝑅-module 𝑄 is called almost approximately nearly quasi compactly 
packed (simply Alappnq compactly packed) if for each family {𝐹𝛼}𝛼∈Ʌ of Alappnq-prime 
submodules of 𝑄 with 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ  there exists 𝛼1,𝛼2,…  ,𝛼𝑛 ∈ Ʌ such that 𝐹 ⊆ ⋃ 𝐹𝛼𝑖.

𝑛
𝑖=1  

Definition 3.2 

An 𝑅-module 𝑄 is called Alappnq compactly packed if every proper submodule of 𝑄 is Alappnq 
compactly packed. 

Remarks and Examples 3.3 

1. 𝑍6 as 𝑍-module Alappnq compactly packed 𝑍-module. 
2. Every module contains a finite number of Alappnq-prime submodules is Alappnq compactly 

packed. 

3. Every proper finite submodule of an 𝑅-module 𝑄 is an Alappnq compactly packed. 

Proposition 3.4 

Let 𝑄 be Alappnq compactly packed 𝑅-module with 𝐽(𝑄) ≠ 𝑄, then 𝑄 satisfies the ascending 
chain condition for Alappnq-prime submodules. 

Proof 

Let 𝐿1 ⊆ 𝐿2 ⊆ 𝐿3 ⊆ ⋯ be ascending chain of Alappnq-prime submodules of 𝑄. Let 𝐿 = ⋃ 𝐿𝑖𝑖 , 
we claim that 𝐿 ≠ 𝑄. In fact if 𝐿 = 𝑄 and 𝐻 is a maximal submodule of 𝑄, then 𝐻 ⊊ ⋃ 𝐿𝑖𝑖 , but 𝑄 
is Alappnq compactly packed module, then there exists 𝛼1,𝛼2,…  ,𝛼𝑛 such that 𝐻 ⊆ ⋃ 𝐿𝛼𝑖

𝑛
𝑖=1  and 

since 𝐿1 ⊆ 𝐿2 ⊆ 𝐿3 ⊆ ⋯ is ascending chain then there exists 𝑞 ∈ {1,2,…  ,𝑛} such that 
⋃ 𝐿𝛼𝑖
𝑛
𝑖=1 = 𝐿𝛼𝑞 then 𝐻 ⊆ 𝐿𝛼𝑞, and since 𝐻 is maximal submodule then 𝐻 = 𝐿𝛼𝑞 and consequently 

𝑄 = ⋃ 𝐿𝑖𝑖 = 𝐿𝛼𝑞 which is a contradiction. So 𝐿 is a proper submodule of 𝑄, thus there exists 

𝛼1,𝛼2,…  ,𝛼𝑛 such that 𝐿 ⊆ ⋃ 𝐿𝛼𝑖
𝑛
𝑖=1 , and since 𝐿1 ⊆ 𝐿2 ⊆ 𝐿3 ⊆ ⋯ is an ascending chain then 

there exists 𝑞 ∈ {1,2,…  ,𝑛} such that ⋃ 𝐿𝛼𝑖
𝑛
𝑖=1 = 𝐿𝛼𝑞 that is ⋃ 𝐿𝑖𝑖 ⊆ 𝐿𝛼𝑞, so𝐿1 ⊆ 𝐿2 ⊆ 𝐿3 ⊆

⋯ ⊆ 𝐿𝛼𝑞. Therefore 𝑄 satisfies the ascending chain condition on Alappnq-prime submodules. 



IHJPAS. 36(1)2023 
 

303 
 

Since every proper submodule of finitely generated module contained in maximal submodule, 

so from the previous proposition, we have the following corollary. 

Corollary 3.5 

If 𝑄 is an Alappnq compactly packed finitely generated module, then 𝑄 satisfies the ascending 
chain condition for Alappnq-prime submodules. 

Also since every proper submodule of multiplication module contained in maximal submodule, 

so from the previous proposition, we have the following corollary. 

Corollary 3.6 

If 𝑄 is Alappnq compactly packed multiplication module, then 𝑄 satisfies the ascending chain 
condition for Alappnq-prime submodules. 

Proposition 3.7 

Let 𝑓:𝑄 → 𝑄′ be an 𝑅-epimorphism, and ker𝑓 is a small submodule of 𝑄, such that ker𝑓 ⊆ 𝑃 
for each Alappnq-prime submodule 𝑃 of 𝑄. Then 𝑄 is an Alappnq compactly packed if and only 
if 𝑄′ is an Alappnq compactly packed. 

Proof 

(⟾) Suppose that 𝑄 is an Alappnq compactly packed 𝑅-module, and 𝐹′ ⊆ ⋃ 𝑃′𝛼𝛼∈Ʌ , where 𝐹′ 
is a proper submodule of 𝑄′ and 𝑃′ is an Alappnq-prime submodule of 𝑄′ for all 𝛼 ∈ Ʌ. Then, 
𝑓−1(𝐹′) ⊆ 𝑓−1(⋃ 𝑃′𝛼𝛼∈Ʌ ) and hence 𝑓

−1(𝐹′) ⊆ ⋃ 𝑓−1(𝑃′𝛼)𝛼∈Ʌ . But by Proposition 2.5 we have 
𝑓−1(𝑃′𝛼) is an Alappnq-prime submodule of 𝑄 for all 𝛼 ∈ Ʌ. Since 𝑄 is an Alappnq compactly 
packed then there exists 𝛼1,𝛼2,…  ,𝛼𝑛 ∈ Ʌ such that 𝑓

−1(𝐹′) ⊆ ⋃ 𝑓−1(𝑃′𝛼𝑖)
𝑛
𝑖=1  implies that 

𝑓−1(𝐹′) ⊆ 𝑓−1(⋃ 𝑃′𝛼𝑖
𝑛
𝑖=1 ). But 𝑓 is an epimorphism then 𝐹′ ⊆ ⋃ 𝑃′𝛼𝑖

𝑛
𝑖=1 . Thus 𝑄

′ is an Alappnq 

compactly packed. 

(⟽) Suppose that 𝑄′ is an Alappnq compactly packed 𝑅-module and ker𝑓 ⊆ 𝑃 for each 
Alappnq-prime submodule 𝑃 of 𝑄. Let 𝐹 be aproper submodule of 𝑄 such that 𝐹 ⊆ ⋃ 𝑃𝛼𝛼∈Ʌ , 
where 𝑃𝛼 is an Alappnq-prime submodule of 𝑄 for all 𝛼 ∈ Ʌ. Then 𝑓(𝐹) ⊆ 𝑓(⋃ 𝑃𝛼)𝛼∈Ʌ  implies 
that 𝑓(𝐹) ⊆ ⋃ 𝑓(𝑃𝛼)𝛼∈Ʌ . But ker𝑓 ⊆ 𝑃𝛼 for each 𝛼. Then by Proposition 2.6 𝑓(𝑃𝛼) is an Alappnq-
prime submodule of 𝑄′ for all 𝛼 ∈ Ʌ. Since 𝑄′ is an Alappnq compactly packed 𝑅-module, then 
there exists 𝛼1,𝛼2,…  ,𝛼𝑛 ∈ Ʌ such that 𝑓(𝐹) ⊆ ⋃ 𝑓(𝑃𝛼𝑖)

𝑛
𝑖=1 . Now, let 𝑥 ∈ 𝐹 then 𝑓(𝑥) ∈ 𝑓(𝐹) ⊆

⋃ 𝑓(𝑃𝛼𝑖)
𝑛
𝑖=1 , then there exists 𝑗 ∈ {1,2,…  ,𝑛} such that 𝑓(𝑥) ∈  𝑓(𝑃𝛼𝑗), implies that there exists 

𝑏 ∈ 𝑃𝛼𝑗 such that 𝑓(𝑥) = 𝑓(𝑏), then 𝑓(𝑥)−𝑓(𝑏) = 0, and 𝑓(𝑥 −𝑏) = 0 so 𝑥 −𝑏 ∈ ker𝑓 ⊆ 𝑃𝛼𝑗. 

That is 𝑥 ∈ 𝑃𝛼𝑗. Hence, 𝐹 ⊆ ⋃ 𝑃𝛼𝑖
𝑛
𝑖=1 , that is 𝐹 is an Alappnq compactly packed submodule. 

Therefore, 𝑄 is an Alappnq compactly packed 𝑅-module. 

The following proposition gives a relation between an Alappnq compactly packed module 𝑄 
and 𝑄𝑆 
Proposition 3.9 

Let 𝑄 be an 𝑅-module, and 𝑆 a multiplicatively closed set in 𝑅. If 𝑄 is an Alappnq compactly 
packed module, then 𝑄𝑆 is an Alappnq compactly packed module. 

Proof 

Let 𝐹 be a proper submodule of 𝑄𝑆, and 𝐹 ⊆ ⋃ 𝑃𝛼𝛼∈Ʌ , where 𝑃𝛼 is an Alappnq-prime submodule 

of 𝑄𝑆 for all 𝛼 ∈ Ʌ. Define 𝑓:𝑄 → 𝑄𝑆 by 𝑓(𝑞) =
𝑞

1
 for every 𝑞 ∈ 𝑄. Thus 𝑓 is an epimorphism. 

Therefore 𝑓−1(𝐹) ⊆ 𝑓−1(⋃ 𝑃𝛼𝛼∈Ʌ ), implies that 𝑓
−1(𝐹) ⊆ ⋃ 𝑓−1(𝑃𝛼)𝛼∈Ʌ . Since 𝑃𝛼 is an 

Alappnq-prime submodule of 𝑄𝑆 for all 𝛼 ∈ Ʌ and 𝑓 is an epimorphism, then by Proposition 2.5 
we have 𝑓−1(𝑃𝛼) is an Alappnq-prime submodule of 𝑄 for all 𝛼 ∈ Ʌ. But 𝑄 is an Alappnq 
compactly packed then there exists 𝛼1,𝛼2,…  ,𝛼𝑛 ∈ Ʌ such that 𝑓

−1(𝐹) ⊆ ⋃ 𝑓−1(𝑃𝛼𝑖)
𝑛
𝑖=1 . Hence, 



IHJPAS. 36(1)2023 
 

304 
 

(𝑓−1(𝐹))𝑆 ⊆ (⋃ 𝑓
−1(𝑃𝛼𝑖)

𝑛
𝑖=1 )𝑆

= ⋃ (𝑓−1(𝑃𝛼𝑖))𝑆
𝑛
𝑖=1 . To prove the last equality, let 

𝑞

𝑠
∈

(⋃ 𝑓−1(𝑃𝛼𝑖)
𝑛
𝑖=1 )𝑆

, where 𝑠 ∈ 𝑆, 𝑞 ∈ ⋃ 𝑓−1(𝑃𝛼𝑖)
𝑛
𝑖=1  so there exists 𝑗 ∈ {1,2,…  ,𝑛} such that 𝑞 ∈

𝑓−1(𝑃𝛼𝑗), thus 
𝑞

𝑠
∈ (𝑓−1(𝑃𝛼𝑗))

𝑆
, hence 

𝑞

𝑠
∈ ⋃ (𝑓−1(𝑃𝛼𝑖))𝑆

𝑛
𝑖=1 . It follows (⋃ 𝑓

−1(𝑃𝛼𝑖)
𝑛
𝑖=1 )𝑆

⊆

⋃ (𝑓−1(𝑃𝛼𝑖))𝑆
 𝑛𝑖=1 . Now, let 

𝑞

𝑠
∈ ⋃ (𝑓−1(𝑃𝛼𝑖))𝑆

𝑛
𝑖=1 , so 

𝑞

𝑠
∈ (𝑓−1(𝑃𝛼𝑗))

𝑆
 for some 𝑗 ∈ {1,2,…  ,𝑛}, 

where 𝑠 ∈ 𝑆, 𝑞 ∈ 𝑓−1(𝑃𝛼𝑗). Hence, 𝑞 ∈ ⋃ 𝑓
−1(𝑃𝛼𝑖)

𝑛
𝑖=1 , thus 

𝑞

𝑠
∈ (⋃ 𝑓−1(𝑃𝛼𝑖)

𝑛
𝑖=1 )𝑆

. Therefore, 

⋃ (𝑓−1(𝑃𝛼𝑖))𝑆
𝑛
𝑖=1 ⊆ (⋃ 𝑓

−1(𝑃𝛼𝑖)
𝑛
𝑖=1 )𝑆

. Now we prove that (𝑓−1(𝐹))𝑆 = 𝐹 for any submodule 𝐹 

of 𝑄𝑆. Let 
𝑥

𝑠
∈ (𝑓−1(𝐹))𝑆 , where 𝑥 ∈ 𝑓

−1(𝐹) and 𝑠 ∈ 𝑆. Then 𝑓(𝑥) ∈ 𝐹, therefore 
𝑥

1
∈ 𝐹, hence 

𝑥

𝑠
=
1

𝑠

𝑥

1
∈ 𝐹. Thus (𝑓−1(𝐹))𝑆 ⊆ 𝐹. Now, let 

𝑥

𝑠
∈ 𝐹, then, 

1

𝑠

𝑥

1
∈ 𝐹 and hence 

𝑥

1
∈ 𝐹, implies that 

𝑓(𝑥) ∈ 𝐹, therefore 𝑥 ∈ 𝑓−1(𝐹) and 
𝑥

𝑠
∈ (𝑓−1(𝐹))𝑆. Thus, 𝐹 ⊆ (𝑓

−1(𝐹))𝑆. Therefore 𝐹 =

(𝑓−1(𝐹))𝑆 for any submodule 𝐹 of 𝑄𝑆. Since (𝑓
−1(𝐹))𝑆 ⊆ ⋃ (𝑓

−1(𝑃𝛼𝑖)) ,𝑆
𝑛
𝑖=1 , we have 𝐹 ⊆

⋃ 𝑃𝛼𝑖
𝑛
𝑖=1 . Hence, 𝐹 is an Alappnq compactly packed submodule of 𝑄𝑆. Thus 𝑄𝑆 is an Alappnq 

compactly packed module. 

4.  Strongly Almost Approximately Nearly Quasi Compactly Packed Modules 

In this section, we introduce the strongly Alappnq compactly packed modules and 

comprehensively study this concept. First, we must introduce the definitions of Alappnq-prime 

radical of submodules, Alappnq-prime radical submodules, and some propositions of these 

concepts needed in the sequel. 

 

Definition 4.1 

Let 𝐹 be proper submodule of an 𝑅-module 𝑄. if there exist an Alappn-prime submodules that 
contain 𝐹, then, the intersection of each Alappn-prime submodules containing 𝐹 is called Alappnq-
prime radical of 𝐹 and denoted by 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹). If there exists no an Alappnq-prime 

submodule containing 𝐹, we put 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝑄. 

Definition 4.2 

We say that a submodule 𝐹 of 𝑄 is Alappnq-prime radical, if 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐹. 

Proposition 4.3 

Let 𝑄 be an 𝑅-module and 𝐹, 𝐿 are submodules of 𝑄. Then: 
1. 𝐹 ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹). 

2. If 𝐹 ⊆ 𝐿, then  𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐿). 

3. 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄 (𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹)) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹). 

Proof 

(1) and (2) direct from definition. 

(3) By part (1) we have 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄 ( 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹)). 

Now 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) =∩𝐾, where the intersection runs over all Alappnq-prime submodules 𝐾 

of 𝑄 with 𝐹 ⊆ 𝐾. 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄 (𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹)) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(∩𝐾) ⊆∩

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐾) =∩𝐾. Hence, 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄 (𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹)) ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹). 

Thus 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄 (𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹)) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹). 



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Proposition 4.4 

Let 𝑄 be an 𝑅-module. If 𝑄 is 𝑍-regular, then 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐹 for all submodule 𝐹 of 𝑄. 

Proof 

Let 𝐹 ⊂ 𝑄. Then by Proposition 4.3(1) we have 𝐹 ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹). Since 𝑄 is 𝑍-regular 

and 𝐹 be aproper submodule of 𝑄 then by Corollary 2.11 we have 𝐹 is the intersection of prime 
submodules. Hence 𝐹 = ⋂ 𝑃𝛼𝛼∈Ʌ  where 𝑃𝛼 is a prime submodule of 𝑄 for each 𝛼 ∈ Ʌ. Therefore 
⋂ 𝑃𝛼𝛼∈Ʌ ⊆ 𝐹, where 𝑃𝛼 is a prime submodule of 𝑄 such that 𝐹 ⊆ 𝑃𝛼. Since by Remark 2.12 every 
prime submodule of 𝑄 is an Alappnq-prime then 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) ⊆ ⋂ 𝑃𝛼𝛼∈Ʌ  implies that 

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐹. 

Proposition 4.5 

Let 𝑄 be an 𝑅-module. If 𝑄 satisfies the ascending chain condition for Alappnq-prime radical 
submodules, then every proper submodule of 𝑄 is an Alappnq-prime radical of a finitely generated 
submodule of it. 

Proof 

Assume that there exists a proper submodule 𝐹 of 𝑄 which is not the Alappnq-prime radical of 
a finitely generated submodule of it. Let 𝑞1 ∈ 𝐹 and 𝐹1 = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞1), so 𝐹1 ⊂ 𝐹. Thus, 

there exists 𝑞2 ∈ 𝐹 −𝐹1. Let 𝐹2 = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞1 +𝑅𝑞2), then, 𝐹1 ⊂ 𝐹2 ⊂ 𝐹, hence there 

exists 𝑞3 ∈ 𝐹 −𝐹3. This implies an ascending chain of Alappnq-prime radical submodules 𝐹1 ⊆
𝐹2 ⊆ 𝐹3 ⊆ ⋯, which does not terminate and this contradicts with hypothesis. 

Now, we introduce the concept of strongly Alappnq compactly packed modules, and study 

some properties. 

Definition 4.6 

A proper submodule 𝐹 of an 𝑅-module 𝑄 is called strongly Alappnq compactly packed if for each 
family {𝐹𝛼}𝛼∈Ʌ of Alappnq-prime submodules of 𝑄 with 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ  there exists 𝛽 ∈ Ʌ such that 
𝐹 ⊆ 𝐹𝛽.  

Definition 4.7 

An 𝑅-module 𝑄 is called strongly Alappnq compactly packed if every proper submodule of 𝑄 
is strongly Alappnq compactly packed. 

Remark 4.8 

Every strongly Alappnq compactly packed submodule is Alappnq compactly packed, but the 

convers is not true as explain in the following example: 

Let 𝑄 = 𝑍2[𝑥] be a module over 𝑍2. Let 𝐿 = {0̅, 1̅,𝑥, 1̅+𝑥} is a 𝑍2-submodule of 𝑄. 𝐿 ⊆ 1̅𝑍2 ∪
𝑥𝑍2 ∪𝑥

2𝑍2 , where 1̅𝑍2,𝑥𝑍2,𝑥
2𝑍2 are prime submodules of 𝑄. But 𝐿 ⊄ 1̅𝑍2 ∪𝑥𝑍2 ∪𝑥

2𝑍2, that 
𝐿 is Alappnq compactly packed, but it is not strongly Alappnq compactly packed. 

The following proposition gives a characterization of strongly Alappnq compactly packed 

modules. 

Proposition 4.9 

Let 𝑄 be an 𝑅-module. Then 𝑄 is strongly Alappnq compactly packed if and only if every 
proper submodule of 𝑄 is Alappnq-prime radical of a cyclic submodule of it. 

Proof 

(⟾) Let 𝐹 be a proper submodule of 𝑄 such that 𝐹 is not Alappnq-prime radical of a cyclic 



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submodule of it, thus for each 𝑞 ∈ 𝐹, 𝐹 ≠ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄((𝑞)). So there exists an Alappnq-prime 

submodule 𝐿𝑞 ⊇ (𝑞) for each 𝑞 ∈ 𝐹 and 𝐹 ⊄ 𝐿𝑞. Thus 𝐹 = ⋃ (𝑞)𝑞∈𝐹 ⊆ ⋃ 𝐿𝑞𝑞∈𝐹 . Since 𝑄 is 

strongly Alappnq compactly packed, then there exists 𝑞0 ∈ 𝐹 such that 𝐹 ⊆ 𝐿𝑞0 which is a 

contradiction. Hence 𝐹 is Alappnq-prime radical of a cyclic submodule of it. 

(⟽) Let 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ , where 𝐹𝛼 is an Alappnq-prime submodule of 𝑄 for all 𝛼 ∈ Ʌ and 𝐹 =

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄((𝑞)) for some 𝑞 ∈ 𝐹. since 𝑞 ∈ 𝐹, thus 𝑞 ∈ ⋃ 𝐹𝛼𝛼∈Ʌ . Hence there exists 𝛽 ∈ Ʌ 

such that 𝑞 ∈ 𝐹𝛽. Thus implies that 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄((𝑞)) ⊆ 𝐹𝛽 and consequently 𝐹 ⊆ 𝐹𝛽 which 

prove that 𝑄 is strongly Alappnq compactly packed. 

The following theorem gives characterizations of strongly Alappnq compactly packed modules. 

Theorem 4.10 

Let 𝑄 be an 𝑅-module. Then the following statements are equivalent: 
1. 𝑄 is strongly Alappnq compactly packed module. 
2. For each 𝐹 ⊂ 𝑄, there exists 𝑞 ∈ 𝐹 such that 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). 

3. For each 𝐹 ⊂ 𝑄, if {𝐹𝛼}𝛼∈Ʌ is a family of submodules of 𝑄, such that 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ , then there 

exists 𝛽 ∈ Ʌ such that 𝐹 ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛽). 

4. For each 𝐹 ⊂ 𝑄, if {𝐹𝛼}𝛼∈Ʌ is a family of Alappnq-prime radical submodule of 𝑄, with 𝐹 ⊆
⋃ 𝐹𝛼𝛼∈Ʌ , then there exists 𝛽 ∈ Ʌ such that 𝐹 ⊆ 𝐹𝛽. 

Proof 

(1) 
        
⇒ (2) Let 𝐹 ⊂ 𝑄. Suppose that 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) ≠ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞) for all 𝑞 ∈ 𝐹. 

implies that for all 𝑞 ∈ 𝐹 there exists an Alappnq-prime submodule 𝐾𝑞 containing 𝑅𝑞 and 𝐹 ⊄ 𝐾𝑞. 

But 𝐹 = ⋃ 𝑅𝑞 ⊆ ⋃ 𝐾𝑞𝑞∈𝐹𝑞∈𝐹  and since 𝑄 is strongly Alappnq compactly packed, then there exists 

𝑞 ∈ 𝐹 such that 𝐹 ⊆ 𝐾𝑞 which is a contradiction. Hence there exists 𝑞 ∈ 𝐹 such that 

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). 

(2) 
        
⇒ (3) Let 𝐹 ⊂ 𝑄, and {𝐹𝛼}𝛼∈Ʌ be a family of submodules of 𝑄, such that 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ . 

Hence, by hypothesis there exists 𝑞 ∈ 𝐹 such that  𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) =  𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). 

Since 𝑞 ∈ 𝐹, then 𝑞 ∈ ⋃ 𝐹𝛼𝛼∈Ʌ  implies that 𝑞 ∈ 𝐹𝛽 for some 𝛽 ∈ Ʌ. Hence, 𝑅𝑞 ⊆ 𝐹𝛽 and 𝐹 ⊆

 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞) ⊆  𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛽). That is  𝐹 ⊆

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛼). 

(3) 
        
⇒ (4) Let 𝐹 ⊂ 𝑄, and {𝐹𝛼}𝛼∈Ʌ be a family of Alappnq-prime radical submodules of 𝑄, such 

that 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ , then by hypothesis there exists 𝛽 ∈ Ʌ such that  𝐹 ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛽) = 𝐹𝛽. 

Since 𝐹𝛽is Alappnq-prime radical submodules of 𝑄. 

(4) 
        
⇒ (1) Let 𝐹 ⊂ 𝑄, and suppose {𝐹𝛼}𝛼∈Ʌ is a family of Alappnq-prime submodules of 𝑄, such 

that 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ . Since 𝐹𝛼 is Alappnq-prime submodules for each 𝛼 ∈ Ʌ then 𝐹𝛼 =
𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛼). Thus 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ = ⋃ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛼)𝛼∈Ʌ . Then by hypothesis, there 

exists 𝛽 ∈ Ʌ such that  𝐹 ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛼) = 𝐹𝛼. Thus 𝑄 is strongly Alappnq compactly 

packed. 

Proposition 4.11 

Let 𝑅 be Alappnq compactly packed ring. Let 𝑄 be a faithful cyclic 𝑅-module such that for 
every submodule 𝐹1,𝐹2 of 𝑄 with 𝐹1 ⊆ 𝐹2, whenever [𝐹1:𝑅 𝑄] ⊆ [𝐹2:𝑅 𝑄]. Then 𝑄 is strongly 
Alappnq compactly packed. 

Proof 

Let 𝐹 be a proper submodule of 𝑄, and {𝐹𝛼}𝛼∈Ʌ is a family of Alappnq-prime submodules of 𝑄, 



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such that 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ . Since 𝑄 is cyclic, then [⋃ 𝐹𝛼𝛼∈Ʌ :𝑅 𝑄] = ⋃ [𝐹𝛼𝛼∈Ʌ :𝑅 𝑄] implies that 
[𝐹:𝑅 𝑄] ⊆ [⋃ 𝐹𝛼𝛼∈Ʌ :𝑅 𝑄] ⊆ ⋃ [𝐹𝛼𝛼∈Ʌ :𝑅 𝑄]. Since 𝑄 is cyclic then 𝑄 is multiplication [12], then by 
Proposition 2.14 [𝐹𝛼:𝑅 𝑄] is Alappnq-prime ideal of 𝑅. But 𝑅 is Alappnq compactly packed ring 
then there exists 𝛼𝑗 ∈ Ʌ such that [𝐹:𝑅 𝑄] ⊆ [𝐹𝛼𝑗:𝑅 𝑄] and by hypothesis, hence 𝐹 ⊆ 𝐹𝛼𝑗. Thus 𝑄 

is strongly Alappnq compactly packed. 

The following proposition gives a necessary and sufficient condition for 𝑍-regular module to 
be strongly Alappnq compactly packed. 

Proposition 4.12 

Let 𝑄 be a 𝑍-regular 𝑅-module, then 𝑄 is strongly Alappnq compactly packed if and only if 
every proper submodule of 𝑄 is cyclic. 

Proof 

(⟾) Suppose that 𝑄 is a strongly Alappnq compactly packed 𝑅-module and let 𝐹 ⊂ 𝑄. Since 
𝑄 is a strongly Alappnq compactly packed, then by Theorem 4.10, there exists 𝑞 ∈ 𝐹 such that 
𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). But 𝑄 is 𝑍-regular module, then by Proposition 4.4, we 

have 𝐹 = 𝑅𝑞, thus 𝐹 is cyclic. 

(⟽) Suppose that every proper submodule of 𝑄 is cyclic. Let 𝐹 be a proper submodule of 𝑄 
then, 𝐹 is cyclic, thus there exists 𝑞 ∈ 𝐹 such that 𝐹 = 𝑅𝑞, so we have 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) =

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). Hence by Theorem 4.10, 𝑄 is strongly Alappnq compactly packed. 

The following proposition gives condition under which strongly Alappnq compactly packed 

module satisfy ascending chain condition on Alappnq-prime radical submodules. 

Proposition 4.13 

Let 𝑄 be strongly Alappnq compactly packed 𝑅-module which has at least one maximal 
submodule, then  𝑄 satisfies the ascending chain condition on Alappnq-prime radical submodules. 

Proof 

Let 𝐹1 ⊆ 𝐹2 ⊆ ⋯ be an ascending chain condition for Alappnq-prime radical submodules of 𝑄, 
let 𝐿 = ⋃ 𝐹𝑖𝑖  then 𝐿 is a submodule of 𝑄. We claim that 𝐿 ⊂ 𝑄. In fact, if 𝐿 = 𝑄 and 𝐻 is a maximal 
submodule of 𝑄, then 𝐻 ⊊ ⋃ 𝐹𝑖𝑖 . Since 𝑄 is strongly Alappnq compactly packed then by Theorem 
4.10 𝐻 ⊆ 𝐹𝑗 for some 𝑗. But 𝐻 is maximal submodule then 𝐻 = 𝐹𝑗 and this implies ⋃ 𝐹𝑖 ⊆ 𝐹𝑗𝑖  that 

is 𝑄 ⊆ 𝐹𝑗 which is a contradiction. So 𝐿 ⊂ 𝑄 and by Theorem 4.10 there exists 𝑗 such that 𝐿 ⊆ 𝐹𝑗, 

so 𝐹1 ⊆ 𝐹2 ⊆ ⋯ ⊆ 𝐹𝑗 that is 𝑄 satisfies the ascending chain condition on Alappnq-prime radical 

submodules. 

The following corollaries are direct consequence of Proposition 4.13. 

Corollary 4.14 

Let 𝑄 be strongly Alappnq compactly packed 𝑅-module such that 𝐽(𝑄) ≠ 𝑄. Then 𝑄 satisfies 
the ascending chain condition on Alappnq-prime radical submodules. 

Corollary 4.15 

If 𝑄 is finitely generated strongly Alappnq compactly packed 𝑅-module, then  𝑄 satisfies the 
ascending chain condition on Alappnq-prime radical submodules. 

Corollary 4.16 

If 𝑄 is multiplication strongly Alappnq compactly packed 𝑅-module, then  𝑄 satisfies the 
ascending chain condition on Alappnq-prime radical submodules. 

In the following proposition we give a condition under which the convers of Proposition 4.13 

is hold. 



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Proposition 4.17 

Let 𝑄 be a Bezout 𝑅-module. If 𝑄 satisfies the ascending chain condition for Alappnq-prime 
radical submodules, then 𝑄 is strongly Alappnq compactly packed module. 

Proof 

Let 𝐹 be a proper submodule of 𝑄, then by Proposition 4.5 there exists a finitely generated 
submodule 𝐿 of 𝐹 such that 𝐹 = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐿) and hence by Proposition 4.3 

𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄 (𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐿)) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐿). But 𝑄 is Bezout 

module then 𝐿 is cyclic submodule, then there exists 𝑞 ∈ 𝐿 such that 𝐿 = 𝑅𝑞, thus implies that 𝑞 ∈
𝐹 and 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). Therefore, by Theorem 4.10 𝑄 is a strongly 

Alappnq compactly packed module. 

Proposition 4.18 

Let 𝑓:𝑄 → 𝑄′ be an 𝑅-epimorphism, and ker𝑓 is a small submodule of 𝑄 such that ker𝑓 ⊆ 𝑃 
for each Alappnq-prime submodule 𝑃 of 𝑄. Then 𝑄 is strongly Alappnq compactly packed if and 
only if 𝑄′ is strongly Alappnq compactly packed. 

Proof 

(⟾) Suppose that 𝑄 is strongly Alappnq compactly packed 𝑅-module, and 𝐹′ ⊆ ⋃ 𝑃′𝛼𝛼∈Ʌ , 
where 𝐹′ is proper submodule of 𝑄′ and 𝑃′ is an Alappnq-prime submodule of 𝑄′ for all 𝛼 ∈ Ʌ. 
Since 𝑓 is an epimorphism, then 𝑓−1(𝐹′) ⊆ 𝑓−1(⋃ 𝑃′𝛼𝛼∈Ʌ ). Thus 𝑓

−1(𝐹′) ⊆ ⋃ 𝑓−1(𝑃′𝛼)𝛼∈Ʌ . But 
𝑃′𝛼 is an Alappnq-prime submodule of 𝑄′, then by Proposition 2.5 we have 𝑓

−1(𝑃′𝛼) is an 
Alappnq-prime submodule of 𝑄 for all 𝛼 ∈ Ʌ. Since 𝑄 is strongly Alappnq compactly packed then 
𝑓−1(𝐹′) ⊆ 𝑓−1(𝑃′𝛽) for some 𝛽 ∈ Ʌ. Therefore 𝐹′ ⊆ 𝑃′𝛽 for some 𝛽 ∈ Ʌ. Hence 𝐹′ is strongly 

Alappnq compactly packed submodule of 𝑄′. Thus 𝑄′ is strongly Alappnq compactly packed. 

(⟽) Suppose that 𝑄′ is strongly Alappnq compactly packed 𝑅-module and ker𝑓 ⊆ 𝑃 for each 
Alappnq-prime submodule 𝑃 of 𝑄. Let 𝐹 be a proper submodule of 𝑄 such that 𝐹 ⊆ ⋃ 𝑃𝛼𝛼∈Ʌ , 
where 𝑃𝛼 is an Alappnq-prime submodule of 𝑄 for all 𝛼 ∈ Ʌ. Then 𝑓(𝐹) ⊆ 𝑓(⋃ 𝑃𝛼)𝛼∈Ʌ  implies 
that 𝑓(𝐹) ⊆ ⋃ 𝑓(𝑃𝛼)𝛼∈Ʌ . But ker𝑓 ⊆ 𝑃𝛼 for each 𝛼. Then by Proposition 2.6 𝑓(𝑃𝛼) is an Alappnq-
prime submodule of 𝑄′ for all 𝛼 ∈ Ʌ. Since 𝑄′ is strongly Alappnq compactly packed 𝑅-module, 
then  𝑓(𝐹) ⊆ 𝑓(𝑃𝛽) for some 𝛽 ∈ Ʌ. Thus, for every 𝑥 ∈ 𝐹, 𝑓(𝑥) ∈ 𝑓(𝐹) ⊆ 𝑓(𝑃𝛽), then 𝑓(𝑥) ∈

 𝑓(𝑃𝛽). Therefore, there exists 𝑏 ∈ 𝑃𝛽 such that 𝑓(𝑥) = 𝑓(𝑏), then 𝑓(𝑥)−𝑓(𝑏) = 0, and 𝑓(𝑥 −

𝑏) = 0 so 𝑥 −𝑏 ∈ ker𝑓 ⊆ 𝑃𝛽. That is 𝑥 ∈ 𝑃𝛽. Therefore 𝐹 ⊆ 𝑃𝛽 for some 𝛽 ∈ Ʌ and hence 𝐹 is 

strongly Alappnq compactly packed submodule. Hence 𝑄 is strongly Alappnq compactly packed 
𝑅-module. 

Proposition 4.19 

Let 𝑄 be an 𝑅-module, and 𝑆 a multiplicatively closed set in 𝑅. If 𝑄 is strongly Alappnq 
compactly packed module, then 𝑄𝑆 is strongly Alappnq compactly packed module. 

Proof 

Assume 𝑄 is strongly Alappnq compactly packed. Let 𝐹 be proper submodule of 𝑄𝑆 with 𝐹 ⊆
⋃ 𝑃𝛼𝛼∈Ʌ , where 𝑃𝛼 is an Alappnq-prime submodule of 𝑄𝑆 for all 𝛼 ∈ Ʌ. Define 𝑓:𝑄 → 𝑄𝑆 by 

𝑓(𝑞) =
𝑞

1
 for every 𝑞 ∈ 𝑄. Thus 𝑓 is an epimorphism. Therefore 𝑓−1(𝐹) ⊆ 𝑓−1(⋃ 𝑃𝛼𝛼∈Ʌ ), implies 

that 𝑓−1(𝐹) ⊆ ⋃ 𝑓−1(𝑃𝛼)𝛼∈Ʌ . Since 𝑃𝛼 is an Alappnq-prime submodule of 𝑄𝑆 for all 𝛼 ∈ Ʌ and 𝑓 
is an epimorphism, then by Proposition 2.5 we have 𝑓−1(𝑃𝛼) is an Alappnq-prime submodule of 
𝑄 for all 𝛼 ∈ Ʌ. But 𝑄 is strongly Alappnq compactly packed, then 𝑓−1(𝐹) ⊆ 𝑓−1(𝑃𝛽) for 

some 𝛽 ∈ Ʌ . Therefore (𝑓−1(𝐹))𝑆 ⊆ (𝑓
−1(𝑃𝛽))𝑆

. We need to show that (𝑓−1(𝐹))𝑆 = 𝐹 for any 

submodule 𝐹 of 𝑄𝑆. Let 
𝑥

𝑠
∈ (𝑓−1(𝐹))𝑆 , where 𝑥 ∈ 𝑓

−1(𝐹) and 𝑠 ∈ 𝑆. Then 𝑓(𝑥) ∈ 𝐹, therefore 



IHJPAS. 36(1)2023 
 

309 
 

𝑥

1
∈ 𝐹, hence 

𝑥

𝑠
=
1

𝑠

𝑥

1
∈ 𝐹. Thus (𝑓−1(𝐹))𝑆 ⊆ 𝐹. Now let 

𝑥

𝑠
∈ 𝐹, then 

1

𝑠

𝑥

1
∈ 𝐹 and hence 

𝑥

1
∈ 𝐹, 

implies that 𝑓(𝑥) ∈ 𝐹, therefore 𝑥 ∈ 𝑓−1(𝐹) and 
𝑥

𝑠
∈ (𝑓−1(𝐹))𝑆. Thus 𝐹 ⊆ (𝑓

−1(𝐹))𝑆. Therefore 

𝐹 = (𝑓−1(𝐹))𝑆 for any submodule 𝐹 of 𝑄𝑆. Now since (𝑓
−1(𝐹))𝑆 ⊆ (𝑓

−1(𝑃𝛽))𝑆
 for some 𝛽 ∈ Ʌ 

we have 𝐹 ⊆ 𝑃𝛽 for some 𝛽 ∈ Ʌ. Thus 𝐹 is strongly Alappnq compactly packed submodule. 

Therefore 𝑄𝑆 is strongly Alappnq compactly packed module. 

5. Conclusion 

The main results of this paper are: 

• Let 𝑄 be Alappnq compactly packed module with 𝐽(𝑄) ≠ 𝑄, then, 𝑄 satisfies the ascending 
chain condition for Alappnq-prime submodules. 

• If 𝑄 is an Alappnq compactly packed module, then, 𝑄𝑆 is an Alappnq compactly packed 
module, for each multiplicatively closed set 𝑆 of 𝑅. 

• An 𝑅-module 𝑄 is strongly Alappnq compactly packed if and only if every proper submodule 
of 𝑄 is Alappnq-prime radical of a cyclic submodule of it. 

• Let 𝑄 be an 𝑅-module. Then the following statements are equivalent: 

1. 𝑄 is strongly Alappnq compactly packed module. 

2. For each 𝐹 ⊂ 𝑄, there exists 𝑞 ∈ 𝐹 such that 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹) = 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝑅𝑞). 

3. For each 𝐹 ⊂ 𝑄, if {𝐹𝛼}𝛼∈Ʌ is a family of submodules of 𝑄, such that 𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ , then 

there exists 𝛽 ∈ Ʌ such that 𝐹 ⊆ 𝐴𝑙𝑎𝑝𝑝𝑛𝑞𝑟𝑎𝑑𝑄(𝐹𝛽). 

4. For each 𝐹 ⊂ 𝑄, if {𝐹𝛼}𝛼∈Ʌ is a family of Alappnq-prime radical submodule of 𝑄, with 
𝐹 ⊆ ⋃ 𝐹𝛼𝛼∈Ʌ , then there exists 𝛽 ∈ Ʌ such that 𝐹 ⊆ 𝐹𝛽. 

 

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