Finitely Generated Module Over  a Discrete Valuation Domain


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(1) (2019), Pages 18-26 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: April 24, 2019 Reviewed: October 18, 2019 Accepted: November 30, 2019 
DOI: http://dx.doi.org/10.18860/ca.v6i1.6798   

A Finitely Generated Module Over  a Discrete Valuation Domain 

Dwi Mifta Mahanani 

Department of Mathematics Brawijaya University 
Email: mdwimifta@yahoo.com 

ABSTRACT 

This article discusses some properties of a finitely generated module over a PID and that over a 
valuation domain which are equivalent. In general, PID is not a valuation domain and also a 
valuation domain is not a PID. But there is a domain which is PID as well as a valuation domain. 
The domain is called a discrete valuation domain (DVR). Therefore, the equivalent properties 
which will be considered are obtained from a finitely generated module over a DVR. Those 
properties are related to a decomposition into direct sum of cyclic submodules and height of an 
element of the module.  

Keywords: DVR; finitely generated module; height of element; PID; valuation domain 

INTRODUCTION 

In general, the properties of an 𝑅 −module depend on the properties of ring 𝑅. For 
example, every finitely generated module over PID can be decomposed into a direct sum 
of cyclic submodules. However, that property does not satisfy in every finitely generated 
modules over a valuation domains. That is to say, not all of those modules can be 
decomposed into a direct sum of cyclic submodules. Moreover, those which can be 
decomposed are those which the Goldie dimension is equal to the length of their series of 
𝑅𝐷 −submodules. The example of finitely generated modules over a valuation domain 
which can not be decomposed can be seen in [5]. 
 In general, a PID is not a valuation domain. An integral domain ℤ, for instance, is a 
PID but not a valuation domain because there are two ideals of ℤ, ℤ2 and ℤ5, which are 
not subset each other. Otherwise, a valuation domain is not a PID.  Note the example 
below. 
 
Example 

Let 𝐾 be a field with transcendental 𝑥, 𝐾𝑖 = 𝐾 [𝑥
1

2𝑖 ] , 𝑃𝑖 = 𝐾 [𝑥
1

2𝑖 ] 𝑥
1

2𝑖  where 𝑖 = 0,1,2, ⋯ . It 

can be seen obviously that 𝐾𝑖  is a PID and 𝑃𝑖  is it’s prime ideal. Moreover, there are 
domains  
 

 𝑂𝑖 = 𝐾𝑖 𝑃𝑖
 

=  {
𝑓(𝑥

1

2𝑖 )

𝑔(𝑥

1

2𝑖 )

|𝑓 (𝑥
1

2𝑖 ) ∈ 𝐾𝑖 , 𝑔 (𝑥
1

2𝑖 ) ∈ 𝐾𝑖 − 𝑃𝑖 }. 

 
 
 
(1) 

http://dx.doi.org/10.18860/ca.v6i1.6798


A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 19 

 
A domain 𝑂 =∪𝑖 𝑂𝑖  is a non-Noetherian valuation domain. Hence 𝑂 is not a PID. The detail 
of the example can be seen in [3] and [6].  
 Meanwhile, there is a domain which is a PID as well as a valuation domain. In [2], 
this domain is called discrete valuation domain (DVR).  Because a DVR is an intersection 
of two different domains, then there must be some properties of finitely generated 
modules over a PID and those over a valuation domain which are equivalent. These 
equivalent properties will be discussed in the result section.  

 

MATERIALS AND METHODS  

The method which is used to write this article is by literature studying on some 
papers and books. There are some properties of a module over a DVR which have been 
discussed in the papers and the books. It will be developed some new properties by 
considering the previous properties. Before we discuss the results, we have to know some 
definitions and properties that we use in the proof of the results. The readers are assumed 
to know about the definition of a module over a ring and some basic properties of a 
module. Therefore, in this article there are no definition and basic properties of a module 
over a ring. 

 
Definition 1 [1, pp. 115] 
Let 𝑀 be a finitely generated 𝑅 −module. The minimal number of generators of 𝑀 is 
denoted by 𝜇(𝑀). If 𝑀 is not finitely generated as 𝑅 −module, then 𝜇(𝑀) = ∞.  
 

Unlike in vector spaces, in modules, not all of those have a basis. For those which 
have a basis are called free modules as which have been explained in [8]. We follow [1] 
that every basis of a free module over a PID  contains 𝜇(𝑀) elements .  
 

There is a submodule of modules over a valuation domain which has an important 
role in those modules. This kind of submodule is an 𝑅𝐷 −submodule. The definition of 
this submodule is presented below. 
 
Definition 2 [4, pp. 38] 
Let 𝑁 be an 𝑅 −submodule of a module 𝑀. This submodule is a relatively divisible 
submodule (𝑅𝐷 −submodule) of 𝑀 if 𝑟𝑁 = 𝑁 ∩ 𝑟𝑀 for all 𝑟 ∈ 𝑅. 
 

There are some properties of 𝑅𝐷 −submodule which will be used to prove the 
equivalence of two theorems in a finitely generated free module over a PID and in that 
over a valuation domain. These properties will be given without any proof.  
 
Lemma 3 [4, pp. 39] 
Let 𝑀 be an 𝑅 −module and 𝐿, 𝑁 are 𝑅 − submodules of 𝑀. Then these properties are 
satisfied. 

1 Direct summand of 𝑀 is an 𝑅𝐷 −submodule of 𝑀. 
2 If 𝐿 ≤ 𝑁 ≤ 𝑀 and 𝐿, 𝑁 are 𝑅𝐷 −submodules of 𝑀, then 𝐿 is an 𝑅𝐷 −submodule of 

𝑁. 
 
Using those properties of 𝑅𝐷 −submodule to proof that a submodule is an 

𝑅𝐷 −submodule is easier than using the definition. The examples related to 



A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 20 

𝑅𝐷 −submodule can be seen below. 
1. Ring  ℤ15 can be assumed to be a ℤ −module. Note that ℤ15 = ℤ3̅ ⊕ ℤ5̅. Then by 

lemma 3, ℤ3̅ and ℤ5̅ are 𝑅𝐷 −submodules of ℤ15. 
2. A domain ℤ can be considered as a module over itself. We know that every 

submodule of this module is an ideal of ℤ. Furthermore, every ideal of ℤ is 
generated by single element (principal ideal). Therefore every two nonzero ideals 
of ℤ always have a nonzero intersection element. Hence, ℤ as a ℤ −module does 
not have any proper nonzero 𝑅𝐷 −submodule.  
 
In a DVR, there is a unique element (up to associates) which has an important role 

to define a height of an element of a module over this domain.  
 

Definition 4 [1, pp.98] 
Let 𝑅 be a ring and 𝑟 ∈ 𝑅. An element 𝑟 is a non nilpotent element of 𝑅 if there is no 
positive integer 𝑛 such that 𝑟𝑛 = 0.  
 

Because of the uniqueness of a non nilpotent in a DVR, we can write every element 
in this domain by a multiplication of a unit element and this non nilpotent element. 
Moreover, this can also be written uniquely. Note the lemma below.  

 
Lemma 5 [7, pp. 5009] 
Let 𝑎 be a nonzero element of a DVR 𝑅. Then 𝑎 can be written uniquely as 𝑎 = 𝑢𝑝𝑘  where 
𝑢 is a unit of 𝑅, 𝑝 is a non nilpotent element of 𝑅 and 𝑘 ∈ ℤ≥0. 
 

The theorem below explains about the decomposition of a finitely generated 
module over a PID into a direct sum of finite cyclic submodules. Moreover, the annihilator 
of  generators of these cyclic submodules  can be arranged such that they form a non 
decreasing chain. Note this theorem below.  
 
Theorem 6 [1, pp. 156] 
Let 𝑀 ≠ 0 be a finitely generated module over a PID 𝑅. If 𝜇(𝑀) = 𝑛, then 𝑀 is isomorphic 

to  

 
 𝑅𝑤1 ⊕ 𝑅𝑤2 ⊕ ⋯ ⊕ 𝑅𝑤𝑛 (2) 

 

such that  

 
 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑤𝑛) ⊆ ⋯ ⊆ 𝐴𝑛𝑛(𝑤2) ⊆ 𝐴𝑛𝑛(𝑤1) ≠ 𝑅. (3) 

 

Moreover, for 1 ≤ 𝑖 ≤ 𝑛,  

 

 
𝐴𝑛𝑛(𝑤𝑖 ) = 𝐴𝑛𝑛 (

𝑀

𝑅𝑤𝑖+1 + ⋯ + 𝑅𝑤𝑛
). 

 
(4) 

 

 
Different from a finitely generated module over a PID, in that over a valuation 

domain, we can find a series of  𝑅𝐷 −submodules. Moreover, the chain can be chosen so 



A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 21 

that the annihilators of the factors forms a non decreasing chain. 
 

Theorem 7 [5, pp. 1798] 
Let 𝑀 ≠ 0 is a finitely generated module over a valuation domain 𝑅. There exist a series 
of 𝑅𝐷 −submodule of 𝑀 
 

 0 = 𝑀0 < 𝑀1 < 𝑀2 < ⋯ < 𝑀𝑛 = 𝑀 
 

(5) 

 
such that  

1. every 𝑀𝑖  is an 𝑅𝐷 −submodule of 𝑀, 
2. every factor module 𝑀𝑖+1/𝑀𝑖  is cyclic.  

Moreover,  it can be chosen that  
 

 
𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑀1) ≤ 𝐴𝑛𝑛 (

𝑀2
𝑀1

) ≤ ⋯ ≤ 𝐴𝑛𝑛 (
𝑀𝑛

𝑀𝑛−1
). 

 

 
(6) 

 
There is a possibility for a finitely generated module over a valuation domain to 

have more than one 𝑅𝐷 −submodules series. Nevertheless, there is another property of 
this module stating that every two 𝑅𝐷 −submodules series are isomorphic. Hence, we can 
choose any series of 𝑅𝐷 −submodule.  

Now consider a finitely generated free module over a PID. In this module, an 
𝑅𝐷 −submodule is the same as a complemented submodule. Note the theorem below.  
 
Theorem 8 [1, pp. 172] 
A submodule of a finitely generated free module over a PID is an 𝑅𝐷 −submodule if and 
only if it is complemented. 
 
 Theorem 8 will be used to construct a direct sum of cyclic submodules of a finitely 
generated free module over a DVR.  
 
Theorem 9 [5, pp. 1801] 
Let 𝑀 be a finitely generated 𝑅 −module where 𝑅 is a valuation domain. Let 0 ≠ 𝑥 ∈ 𝑀 be 
any element of 𝑀. There exist 𝑟 ∈ 𝑅 such that 𝑥 ∈ 𝑟𝑀 − 𝑟𝑃𝑀 where 𝑃 is a maximal ideal 
of 𝑅. 
 

The theorem above ensures the existence of an element 𝑟 ∈ 𝑅 which has a property 
that 𝑥 always in 𝑟𝑀 but not in 𝑟𝑃𝑀. If there is another element 𝑠 ∈ 𝑅 which satisfies the 
property in the theorem 9, then the ideal 𝑅𝑟 will be equal to ideal 𝑅𝑠. Theorem 9 and that 
argument motivate a definition of height of an element in a module over a valuation 
domain. The definition is presented below.  
 
Definition 10 [5, pp. 1802] 
Let 𝑀 be a finitely generated 𝑅 −module where 𝑅 is a valuation domain. Let 0 ≠ 𝑥 ∈ 𝑀 be 
any element of 𝑀. Height of 𝑥, which is denoted by ℎ𝑀(𝑥), is defined as an ideal 𝑅𝑟 such 
that 𝑥 ∈ (𝑟𝑀 − 𝑟𝑃𝑀) where 𝑃 is a maximal ideal of 𝑅. Height of 0 is defined as a zero ideal. 
 

The definition of height of an element of a finitely generated module over a 



A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 22 

valuation domain and that over a DVR are different. In a module over a valuation domain, 
the height of an element is in the form of an ideal. However, in a module over a DVR, it is 
a non negative integer. In fact, DVR is a special case of valuation domain. Those two 
statements actually are equivalent. This equivalence will be proven in the next theorem. 
But before doing the proof, note the definition of height of an element of a module over a 
DVR.  
 
Definition 11 [7, pp. 5033] 

Let 𝑎 ∈ 𝑀 be any nonzero element where 𝑀 is an 𝑅 −module over a DVR. Height of 𝑎 is the 
greatest non negative integer 𝑛 which equation 𝑝𝑛𝑥 = 𝑎 is soluble. If there is no such integer, 
it means the height is ∞. Element 𝑝 in the equation is a non nilpotent element in 𝑅. 
 

From the definition above, the height of zero element in 𝑀 is ∞. It is because there 
is no greatest non negative integer 𝑛 such that 𝑝𝑛𝑥 = 0 is soluble.  

RESULTS AND DISCUSSION  

We know that a DVR is a Noetherian valuation domain.  Therefore, every property 
in a module over a valuation domain always satisfies in that over a DVR. One of  those 
properties is the existence of a series of RD-submodules. Note an illustration below.  

Let 𝑀 be a finitely generated 𝑅 −module over a PID and 𝜇(𝑀) = 𝑛. It will be 

constructed a series of 𝑅𝐷 −submodule of 𝑀. Based on theorem 6, module 𝑀 is 

isomorphic to  

 𝑅𝑤1 ⊕ 𝑅𝑤2 ⊕ ⋯ ⊕ 𝑅𝑤𝑛. 

 

(7) 

With this direct sum, we can make a series. Observe this series below. 

 

 0 = 𝑀0 < 𝑅𝑤𝑛 < 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 < ⋯ < 𝑅𝑤𝑛 ⊕ ⋯ ⊕ 𝑅𝑤2 ⊕ 𝑅𝑤1 = 𝑀. 

 

(8) 

Based on lemma 3, submodule 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘  where 1 ≤ 𝑘 ≤ 𝑛 is an 

𝑅𝐷 −submodule of 𝑀. Hence, there exist a series of 𝑅𝐷 −submodule of 𝑀.  

Observe that 

 
(

𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1
𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘

) 

 

 
(9) 

is cyclic with the generator 𝑅(𝑤𝑘−1 + (𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 )) for 1 ≤ 𝑙 ≤ 𝑛 − 1.  

Let 

 

 
𝑟 ∈ 𝐴𝑛𝑛 (

𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1
𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘

). 

 

 
(10) 

Then 𝑟𝑤𝑘−1 ∈ (𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ).  Therefore,  

 

  𝑟𝑤𝑘−1 ∈ 𝑅𝑤𝑘−1 ∩ (𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ). 

 

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A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 23 

Because 𝑅𝑤𝑘−1 + 𝑅𝑤𝑘 + ⋯ + 𝑅𝑤𝑛 is a direct sum, then it causes 𝑟𝑤𝑘−1 = 0. It means that 

𝑟 ∈ 𝐴𝑛𝑛(𝑤𝑘−1). Based on theorem 6,  𝐴𝑛𝑛(𝑤𝑘−1) = 𝐴𝑛𝑛 (
𝑀

𝑅𝑤𝑘+⋯+𝑅𝑤𝑛
).  It is obtained that 

𝑟𝑤𝑘−2 = 𝑟𝑘 𝑤𝑘 + ⋯ + 𝑟𝑛𝑤𝑛 for some 𝑟𝑖 ∈ 𝑅. Note that  

 

 𝑟𝑤𝑘−2 + (𝑅𝑤𝑛 + ⋯ + 𝑅𝑤𝑘−1) =  (𝑟𝑘 𝑤𝑘 + ⋯ + 𝑟𝑛𝑤𝑛) + (𝑅𝑤𝑛 + ⋯ + 𝑅𝑤𝑘−1) 

= 𝑅𝑤𝑛 + ⋯ + 𝑅𝑤𝑘−1. 

 

 
(12) 

Hence 

 

 
𝑟 ∈ 𝐴𝑛𝑛 (

𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−2
𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1

). 

 

 
(13) 

Therefore, it is proved that  

 

 
𝐴𝑛𝑛 (

𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1
𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘

)

≤ 𝐴𝑛𝑛 (
𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−2
𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1

). 

 
 
 
(14) 

 

Now, let 0 ≠ 𝑀 is a finitely generated free module over a DVR 𝑅. Based on theorem 

7, there exist a series of  𝑅𝐷 −submodule  

 

 0 = 𝑀0 < 𝑀1 < 𝑀2 < ⋯ < 𝑀𝑛 = 𝑀 

 

(15) 

with every factor module is cyclic. Note that 𝑀𝑖  is an 𝑅𝐷 −submodule of 𝑀𝑖+1 because 𝑀𝑖  

is an 𝑅𝐷 −submodule of 𝑀 for all 𝑖. Based on theorem 8 𝑀𝑖  is complemented in 𝑀𝑖+1. 

Therefore it can be written  

 

 𝑀𝑖+1 = 𝑀𝑖 ⊕ 𝑁𝑖+1. (16) 
 

Note that 𝑀𝑖+1/𝑀𝑖 ≅ 𝑁𝑖+1 and 𝑀𝑖+1/𝑀𝑖  is cyclic for every 𝑖. Hence 𝑁𝑖+1 is cyclic. It can be 

written 𝑁𝑖+1 = 𝑅𝑎𝑖+1. Therefore, it is obtained  

 

 𝑀 = 𝑀𝑛−1 ⊕ 𝑅𝑎𝑛 

= 𝑀𝑛−2 ⊕ 𝑅𝑎𝑛−1 ⊕ 𝑅𝑎𝑛 

⋮ 

= 𝑅𝑎1 ⊕ 𝑅𝑎2 ⊕ ⋯ ⊕ 𝑅𝑎𝑛. 

 

 
 
 
(17) 

Now consider that 𝐴𝑛𝑛 (
𝑀𝑖+1

𝑀𝑖
) = 𝐴𝑛𝑛(𝑁𝑖+1) = 𝐴𝑛𝑛(𝑎𝑖+1).  If we choose the series 

of 𝑅𝐷 −submodule so that 

 

 
𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑀1) ≤ 𝐴𝑛𝑛 (

𝑀2
𝑀1

) ≤ ⋯ ≤ 𝐴𝑛𝑛 (
𝑀𝑛

𝑀𝑛−1
), 

(18) 



A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 24 

 
 then we have a chain  

 

 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑎1) ≤ 𝐴𝑛𝑛(𝑎2) ≤ ⋯ ≤ 𝐴𝑛𝑛(𝑎𝑛). 

 

(19) 

By renumbering, that is 𝑎𝑛−𝑖 = 𝑤𝑖+1, we have a chain  

 

 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑤𝑛) ≤ 𝐴𝑛𝑛(𝑤𝑛−1) ≤ ⋯ ≤ 𝐴𝑛𝑛(𝑤1). 

 

(20) 

Illustration above shows that Theorem 6 and Theorem 7 are equivalent if 𝑀 is a finitely 

generated free module over a DVR.  

 

Theorem 12 

In a finitely generated free module over a DVR, theorem 6 and theorem 7 are equivalent. 

Proof.  

∎ 

 

Beside the equivalent statement, there are also some properties of the module 

which related to the property of the domain. In this case, we will assume the domain is a 

valuation domain.  As we know, the properties will also satisfy if the domain is a DVR. The 

comparable property of ideals of a valuation domain is inherited to submodules of a cyclic 

module over a valuation domain. Note the lemma below. 

 

Lemma 13 

Let 0 ≠ 𝑅𝑚 is a cyclic module over a valuation domain 𝑅. If 𝑁 and 𝑁′ are submodules of 

𝑅𝑚, then 𝑁 ⊆ 𝑁′ or 𝑁′ ⊆ 𝑁. 

Proof. 

Let 𝐼 = {𝑎 ∈ 𝑅|𝑎𝑚 ∈ 𝑁} and 𝐽 = {𝑏 ∈ 𝑅|𝑏𝑚 ∈ 𝑁′}. These two sets are ideals of 𝑅. 

We know that 𝑅 is a valuation domain. Hence we have 𝐼 ⊆ 𝐽 or 𝐽 ⊆ 𝐼. Let 𝑥 = 𝑎𝑚 ∈ 𝑁 be 

any element. Then 𝑎 ∈ 𝐼. If 𝐼 ⊆ 𝐽, then 𝑎𝑚 ∈ 𝑁′. Therefore, 𝑁 ⊆ 𝑁′. We can prove 

analogously for 𝑁′ ⊆ 𝑁. 

∎ 

By lemma 13, a cyclic module over a valuation domain can not be decomposed into 

a direct sum of its nonzero submodules. Hence a cyclic module over a valuation domain is 

an indecomposable module. From this fact, we have this corollary below.  

 

Corollary 14 

If 𝑀 is a finitely generated module over a DVR 𝑅, then 𝑀 is isomorphic to direct sum of 

cyclic indecomposable submodules.  

Proof. 

Use theorem 6 and lemma 13. 

∎ 

 



A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 25 

Lemma 15 

Non nilpotent element of a DVR 𝑅 is a prime element.  

Proof. 

Let 𝑝 be a non nilpotent element of 𝑅 and 𝑝|(𝑎𝑏) where 𝑎, 𝑏 ∈ 𝑅. To show that 𝑝 is 

a prime element of 𝑅, we need to show that 𝑝|𝑎 or 𝑝|𝑏. Based on lemma 5, let 𝑎 = 𝑢1𝑝
𝑛1  

and 𝑏 = 𝑢2𝑝
𝑛2  with 𝑢1, 𝑢2 are unit elements of 𝑅 and 𝑛1 ∈ ℤ>0 or 𝑛2 ∈ ℤ>0. Then it is 

obvious that 𝑝|𝑢1𝑝
𝑛1  or 𝑝|𝑢2𝑝

𝑛2 . Hence 𝑝 is a prime element of 𝑅. 

∎ 

 

If there is another prime element of 𝑅, namely 𝑞, then every element of 𝑅 can also 

be written in the form of 𝑢𝑞𝑛 uniquely where 𝑢 is a unit of 𝑅 and 𝑛 ∈ ℤ≥0. Note this 

explanation below.  

If 𝑞 is another prime element of 𝑅, then 𝑅𝑞 is a maximal ideal of 𝑅. The uniqueness 

of maximal ideal of 𝑅 makes 𝑅𝑝 = 𝑅𝑞. Hence we have 𝑝 = 𝑟𝑞 for some 𝑟 ∈ 𝑅. It means 

𝑝|𝑟𝑞. Because 𝑝 is prime element, then 𝑝|𝑟 or 𝑝|𝑞.  

If 𝑝|𝑟, then 𝑝𝑠 = 𝑟 for some 𝑠 ∈ 𝑅. We have 𝑝 = 𝑝𝑠𝑞. It means 𝑠𝑞 = 1. Therefore 𝑞 

is a unit element of 𝑅. It is contradiction because 𝑞 is prime element. Therefore, we have 

𝑝|𝑞. Let 𝑝𝑡 = 𝑞 for some 𝑡 ∈ 𝑅. Then we have 𝑝 = 𝑟𝑡𝑝. It means 𝑟𝑡 = 1. Hence, 𝑟 is a unit 

of 𝑅. Therefore, 𝑝 and 𝑞 are associate.  

Definition 10 is defined in a finitely generated module over a valuation domain. It 

means the definition satisfies also for that over a DVR. Hence the definition 10 and 

definition 11 must be equivalent for a DVR. But before proving this equivalence, we need 

the property below. 

 

Lemma 16 

Let 𝑅 be a DVR and 𝑎 is a non nilpotent element of 𝑅. Then 𝑅𝑎 is a maximal ideal of 𝑅.   

Proof.  

By lemma 15, 𝑎 is a prime element of 𝑅. Then 𝑅𝑎 is a prime ideal which is a maximal ideal 

of 𝑅. 

∎ 

 

Theorem 17 

Definition 10 and definition 11 are equivalent in a finitely generated module over a DVR. 

Proof. 

Let 𝑃 be a maximal ideal of 𝑅 and 𝑎 be any nonzero element of 𝑀. Because 𝑅 is a 

DVR, then 𝑃 = 𝑅𝑝 for 𝑝 is a non nilpotent element of 𝑅. Based on definition 10 𝑎 ∈ 𝑟𝑀 −

𝑟𝑃𝑀 for some 𝑟 ∈ 𝑅. We can write 𝑎 = 𝑟𝑚 = 𝑢𝑝𝑛 𝑚 with 𝑢 is a unit of 𝑅, 𝑛 ∈ ℤ≥0, 𝑚 ∈ 𝑀 

and 𝑟 = 𝑢𝑝𝑛. Because 𝑎 ∉ 𝑟𝑃𝑀 = 𝑟𝑝𝑀, then 𝑎 ∉ 𝑝𝑛+1𝑀. Hence 𝑛 is the greatest non 

negative integer such that 𝑝𝑛𝑥 = 𝑎 have a solution.  

Now let 𝑛 be the greatest non negative integer such that 𝑝𝑛𝑥 = 𝑎 is soluble. 

It means that 𝑎 ∈ 𝑝𝑛𝑀 − 𝑝𝑛+1𝑀. Hence based on definition 10, the height of 𝑎 is 𝑅𝑝𝑛.  

∎ 



A Finitely Generated Module over a Discrete Valuation Domain  

Dwi Mifta Mahanani 26 

CONCLUSIONS 

In a finitely generated free module over a DVR, we can make a series of it’s 
𝑅𝐷 −submodules by arranging it’s cyclic submodules which are the summand of it’s 
decomposition. Furthermore, from this series, we also get a special series which has a 
property that it’s sequence of the annihilator of it’s factor is ordered non-decreasingly. 
Otherwise, if we have a series of 𝑅𝐷 −submodule, we can decompose the module into a 
direct sum of cyclic submodules. Moreover, these cyclic submodules are indecomposable 
submodules.  Besides, the height of an element of a finitely generated module over a DVR 
can be considered as an ideal of it’s domain or a non negative integer which is the prime 
power of a generator of the ideal. 
 

REFERENCES 

 

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[3] Knaf, H. (https://math.stackexchange.com/users/2479/hagen-knaf), "Examples of 
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https://math.stackexchange.com/q/363201. 

[4] Laszlo, F. and Luigi, S, "Modules over Non-Noetherian Domains", Rhode Island: 
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[5] Luigi, S. and Paolo, Z., "Finitely Generated Modules over Valuation Ring". Comm. 
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[6] Mahanani, D.M., "Thesis: Karakterisasi Dekomposisi Modul yang Dibangun secara 
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[7]  Piotr K., Askar A.T., "Modules over Discrete Valuation Domains I", Journal of 
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[8] Roman, S., "Advanced Linear Algebra  Third Edition", Springer, 2000. 

 

 

 

 

 
 
 

 
 
 

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