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THAI NGUYEN UNIVERSITY
THAI NGUYEN UNIVERSITY OF SCIENCES

TRAN DUC DUNG

ON THE SEQUENTIAL POLYNOMIAL TYPE
AND REDUCIBILITY INDEX OF MODULE ON
COMMUTATIVE RINGS

SUMMARY OF MATHEMATICS DOCTOR THESIS

Thai Nguyen - 2019


THAI NGUYEN UNIVERSITY
THAI NGUYEN UNIVERSITY OF SCIENCES

TRAN DUC DUNG

ON THE SEQUENTIAL POLYNOMIAL TYPE
AND REDUCIBILITY INDEX OF MODULE ON
COMMUTATIVE RINGS

Major: Algebra and number theory
Code: 9 46 01 04

SUMMARY OF MATHEMATICS DOCTOR THESIS

Suppervisors:
Prof. Dr. Sc. Nguyen Tu Cuong
Prof. Dr. Le Thi Thanh Nhan

Thai Nguyen - 2019


1

Preliminaries

Let (R, m) be a Noetherian local ring and M a finitely generated
R-module with dim M = d. We have depth M ≤ dim M . M is a CohenMacaulay if depth M = dim M . Cohen-Macaulay module plays a central
role in Commutative Algebra and appears in many different areas of study
of Mathematics such as Algebraic Geomery, Combined Theory, Invariant
Theory...
Note that M is Cohen-Macaulay if and only if (M/xM ) = e(x; M )
for every parameter system x of M . One of the important extensions of the
Cohen-Macaulay module class is the Buchsbaum module class introdeced
by J. St¨
uckrad and W. Vogel, that is the class of module M satisfy the hypothesis by D.A. Buchsbaum: (M/xM )−e(x; M ) is constant independent
of parameter system x. Then, N.T. Cng, P. Schenzel and N.V. Trung has
introduced a class os M module satisfactory supx ( (M/xM ) − e(x; M )) <
∞, is called generalized Cohen-Macaulay module. In 1992, N.T. Cuong
introduced an invariant p(M ) of M , called the polynomial type of M , in
order to measure the non-Cohen-Macaulayness of M , thereby classifying
and study structure of a finitely module over a local ring. If we stipulate
the degree of zero polynomial to be −1, then M is Cohen-Macaulay if and
only if p(M ) = −1, an M is generalized Cohen-Macaulay if and only if
p(M ) ≤ 0.


2

An important generalized of the notion of Cohen-Macaulay module
is that of sequentially Cohen-Macaulay module, introduced almost at the
same time by R.P. Stanley in the graded setting and by P. Schenzel in
the local setting: M is said to be sequentially Cohen-Macaulay if the quotient module Di /Di+1 is Cohen-Macaulay, where D0 = M and Di+1 is the
largest submodule of M of dimension less than dim Di for all i ≥ 0. Then,
N.T. Cuong and L.T. Nhan introduced the sequential generalized CohenMacaulay is defined similarly to the sequential Cohen-Macaulayness except
that each quotient module Di /Di+1 is required to be generalized CohenMacaulay instead of being Cohen-Macaulay.
The first purpose of thesis is introduce the notion of sequential polynomial type of M , which is denote by sp(M ), in order to measure how far
M is different from the sequential Cohen-Macaulayness. We showed that
sp(M ) is dimension of the non sequentially Cohen-Macaulay locus of M if
R is a quotient of Cohen-Macaulay local ring. We study change of the sequential polynomial type under localization, m-adic and an ascent-descent
property of sequential polynomial type between M and M/xM for certain
parameter x of M . We describe sp(M ) in term of the deficiency modules
of M when R is a quotient of a Gorenstein local ring. Note that N.T.
Cuong, D.T. Cuong v H.L. Truong studied a new invariant of M through
multiplicity, and ring R is a quotient of a Cohen-Macaulay local ring then
this invariant is the sequential polynomial type of M . Recently, S. Goto
v L.T. Nhan (2018) showed a parameter characteristics of the sequential
polynomial type.
The second purpose of thesis is research some problems about reducibility index of finitely generated module on local ring. A submodule
N of M is called an irreducible submodule if N can not be written as an
intersection of two properly larger submodules of M . The number of ir-


3

reducible components of an irredundant irreducible decomposition of N ,
which is independence of the choice of the decompostion by E.Noether,
is called the index of reducibility of N and denoted by irM (N ). If q is a

parameter ideal of M , then irM (qM ) is said to be the index of reducibility
of q on M .
A uniform bound for index of reducibility of parameter for CohenMacaulay class, Buchsbaum class, generalized Cohen-Macaulay class has
been resaarched by many mathematicians. Recently, P.H. Quy (2013)
showed a uniform bound of irM (qM ) for all parameter ideals q of M in
the case where p(M ) ≤ 1. In case where p(M ) ≥ 3, a uniform bound
of irM (qM ) for all parameter ideals q of M may not exist even when
sp(M ) = −1. In fact, Goto and Suzuki (1984) constructed a sequentially
Cohen-Macaulay Noetherian local ring (R, m) such that p(R) = 3 and the
supremum of irR (q) is infinite, where q runs over all parameter ideals of R.
On the other hand, the notion of good parameter ideal (such ideals exist)
makes an important role in the study of modules which are not necessarily
unmixed. Therefore, it is natural to ask if there exists a uniform bound
of irM (qM ) for all good parameter ideals q of M . Some positive answers
are given by H. L. Truong (2013) for the case where sp(M ) = −1, and by
P.H. Quy (2012) for the case where sp(M ) ≤ 0. In this thesis, we study
a uniform bound of irM (qM ) for all good parameter ideals q of M where
sp(M ) ≤ 1. On the other hand, we study the reducibility index of Artian
module and clarify the relationship between irM (N ) and irR (D(M/N )),
where irR (D(M/N )) is the sum-reducibility index of the Matlis dual of
M/N , that is the number of sum-irreducible submodules appearing in an
irredundant sum-irreducible representation of D(M/N ). This is a very
basic problem that was first studied in this thesis.
Regarding the approach, to study the sequential polynomial type we


4

exploit the properties of the dimension filtration of the module (dimension
filtration concept introduced by P. Schenzel and adjusted by NT Cuong

and LT Nhan for convenience for use), the strict filter regular sequence
introduced by N.T. Cuong, M. Morales and L.T. Nhan and peculiar properties of the Artin module, especially the local cohomology module with
respect to m. In order to study a uniform bound of the index of good
parameters when sp(M ) is small, we use the theory of good parameter system introduced by N.T. Cuong, .T. Cuong, characteristic of homogeneity
of the sequential polynomial type and the results of J.D. Sally about the
minimal number of generators of the module.
The thesis is divided into 3 chapters. Chapter 1 reiterates some basic knowledge of commutative algebra in order to base on presenting the
main content of the thesis in the following chapters, including: local cohomology with respect to maximal ideal, secondary representation of the
Artinian module, polynomial type, Cohen-Macaulay module, generalized
Cohen-Macaulay module, sequentially Cohen-Macaulay module, sequentially generalized Cohen-Macaulay module.
Chapter 2 presents the sequential polynomial type of the module.
Section 2.1 shows the relationship between dimensional filtration of M and
dimensional filtration of M/xM , where x is a strict regular filter element
(Prosition 2.1.8). Section 2.2 introduces the concept of the sequential polynomial type of M, denoted by sp(M ) to measure the non-sequential-CohenMacaulayness of M. Proposition 2.2.4 provides the relationship between
sp(M ) and the dimension of non-sequentially Cohen-Macaulay locus of M .
Next, we give information about the sequential polynomial type under localization and m-aic completion (Theorem 2.2.7, Theorem 2.2.9). Section
2.3 provides the relationship between sp(M/xM ) and sp(M ) , where x is a
certain parameter element (Theorem 2.3.4). The main result of the chap-


5

ter (Section 2.4) provides homogeneous characteristics of the sequential
polynomial type (Theorem 2.4.2).
Chapter 3 presents some problems of the reducibility index of module. Section 3.2 proof the existence of a uniform bound of good parameter
parameters q of M with sp(M ) ≤ 1 (Theorem 3.2.6). Section 3.3 investigates the index of reducibility in the Artinian module category and gives
a comparison between the index of the submodule of M with the index
of Matlis duality of the corresponding quotient module of M (Theorem
3.3.10).

6

CHAPTER 1
Preparation knowledge
In this chapter, we recall some basic knowledge of commutative algebra in
order to base on presenting the main content of the thesis in the following chapters, including: local cohomology with respect to maximal ideal,
secondary representation of the Artinian module, polynomial type, CohenMacaulay module and its extensions.
The notion of the polynomial type p(M ) was introduced by N.T.
Cuong (1992), in order to measure how far the module M is from belonging
to the class of Cohen-Macaulay modules. For each system of parameters
x = (x1 , . . . , xd ) of M and each tuple of d positive integers n = (n1 , . . . , nd ),
we consider the difference
IM,x (n) =

n1
nd
R (M/(x1 , . . . , xd )M )

− n1 . . . nd .e(x, M )

as a function in n1 , . . . , nd , where e(x, M ) denotes the multiplicity of M
with respect to x. In general, IM,x (n) is not a polynomial for n1 , . . . , nd
large enough, but it takes non-negative values and it is bounded above by
polynomials.
Definition 1.2.1. The least degree of all polynomials bounding
above the function IM,x (n), which does not depend on the choice of x, is
called the polynomial type of M and denoted by p(M ).

7

The concept of dimensional filtration is introduced by P. Schenzel
(1998). Then N.T. Cuong and L.T. Nhan (2003) has slightly adjusted this
definition by removing repeating components to make it more convenient
for use.
Definition 1.3.1. A filtration Hm0 (M ) = Dt ⊂ . . . ⊂ D1 ⊂ D0 =
M of submodules of M is said to be the dimesion filtration of M , if for
each 1 ≤ i ≤ t, Di is the largest submodule of M of dimension less than
dimR Di −1 .
The notion of sequentially Cohen-Macaulay module was introduced
by R. Stanley in the graded setting and by P. Schenzel in the local setting.
This notion was extended to the concept of sequentially generalized CohenMacaulay module in a natural way.
Definition 1.3.2. Let Hm0 (M ) = Dt ⊂ . . . ⊂ D1 ⊂ D0 = M be
the dimension filtration of M . The module M is said to be sequentially
Cohen-Macaulay if each quotient module Di−1 /Di is Cohen-Macaulay. If
Di−1 /Di is generalized Cohen-Macaulay for all i = 1, . . . , t, then M is said
to be sequentially generalized Cohen-Macaulay.
The concept of good parameter system introduced by N.T. Cuong
and .T. Cuong to study the class of Cohen-Macaulay modules and its
extension.
Definition 1.3.3. A filtration M = H0 ⊃ H1 ⊃ . . . ⊃ Hn of
submodules of M is said to satisfy the dimension condition if dim R Hi <
dim R Hi−1 for all i ≤ n. A parameter ideal q = (x1 , . . . , xd ) of M is said to
be a good parameter ideal with respect to such a filtration if (xhi +1 , . . . , xd )M ∩
Hi = 0 for all i ≤ n, where hi = dim R Hi . If q is good with respect to the
dimension filtration, then it is simply called a good parameter ideal of M .


8

CHAPTER 2
The sequential polynomial type of
module
Through this chapter, let (R, m) be a Noether local ring and M a
finitely generated R-modulem with dim M = d, A be an R-module Artinian. We denoted R and M the m-adic completion of R and M respectively.

2.1

Dimension filtration and strict filter regular sequence
By N.T. Cuong, M. Morales and L.T. Nhan, an element x ∈ m is called

strict filter regular element (strict f-element for short) of M if x ∈
/ p vi mi
p∈(

AttR Hmj (M )) \ {m}. The main result in this section is to show the

j≤d

relationship between dimension filtration of M and dimension filtration of
M/xM , where x is strict f-element of Di−1 /Di for every i = 1, ..., t.
Prosition 2.1.8. Suppose that R is a quotient of Cohen-Macaulay local
ring. Let Hm0 (M ) = Dt ⊂ ... ⊂ D1 ⊂ D0 = M is dimension filtration
of M and x ∈ m be a strict f -element of Di−1 /Di for all i ≤ t. Set
Di = (Di +xM )/xM for i ≤ t. Let Hm0 (M/xM ) = Lt ⊂ ... ⊂ L0 = M/xM


9

is dimension of M/xM . Then we have
(i) t ≤ t ≤ t + 1. Concretely, t = t if dt−1 ≥ 2 and t = t + 1 if dt−1 = 1.
(ii) Di ⊆ Li v (Li /Di ) < ∞ for all i ≤ t .

2.2

Sequentialy polynomial type: Localization and
completion
Throughout this section, let Hm0 (M ) = Dt ⊂ . . . ⊂ D1 ⊂ D0 = M be

the dimension filtration of M and di := dim Di for all i ≤ t.
Definition 2.2.1 The sequential polynomial type of M , denoted by sp(M )
is defined throghout by the polynomial type of a quotient module in dimension filtration of M :
sp(M ) = max{p(Di−1 /Di ) | i = 1, . . . , t}.
It is clear that sp(M ) = −1 if and only if M is sequentially CohenMacaulay. Moreover, sp(M ) ≤ 0 if and only if M is sequentially generalized
Cohen-Macaulay. In general, sp(M ) measures how far M is different from
the sequential Cohen-Macaulayness. Let
nSCM(M ) := {p ∈ Spec(R) | Mp is not a sequentially Cohen-Macaulay Rp -module}
denote the non sequentially Cohen-Macaulay locus of M .
We have the following relation between sp(M ) and the dimension of the
non-sequentially Cohen-Macaulay locus of M .
Prosition 2.2.4. If R is catenary then sp(M ) ≥ dim(nSCM(M )). The
equality holds true if R is a quotient of local Cohen-Macaulay.
Next we study the sequential polynomial type under localization.
Theorm 2.2.7. Let p ∈ SuppR M . Suppose that R is catenary.
(i) If dim(R/p) > sp(M ) then Mp is a sequentially Cohen-Macaulay Rp -


10

module.
(ii) If dim(R/p) ≤ sp(M ) then sp(Mp ) ≤ sp(M ) − dim(R/p).
Note that p(M ) = p(M ), however we do not have such a relationship
between sp(M ) and sp(M ).
Example 2.2.8. Let (R, m) be the Noetherian local domain of dimension
2 constructed by D. Ferrand and M. Raynaud such that R has an embedded
associated prime P of dimension 1. Then sp(R) = 1 but sp(R) = −1.
Following M. Nagata, R is called unmixed if dim R/p = dim R for
every p ∈ Ass R. The following results show the relationship between
sp(M ) and sp(M ), at the same time we give a criterion for sp(M ) and
sp(M ) to be the same.
Theorem 2.2.9.

sp(M ) ≤ sp(M ). The equality holds true if R/p is

unmixed for all associated primes p of M.
Without the unmixedness of associated primes, sp(M ) and sp(M )
may be different.
Example 2.2.10. For any integer r ≥ 0, there exists a Noether local
domain (R∗ , m∗ ) which is university catenary such that sp(R∗ ) = −1 and
sp(R∗ ) = r + 2.

2.3

A relation between sp(M ) and sp(M/xM ) where x
is a parameter element
In this section, we show a relation between sp(M ) and sp(M/xM ),

where x is a certain parameter of M . Let Hm0 (M ) = Dt ⊂ ... ⊂ D0 = M
be the dimension filtration of M and di := dim Di for all i ≤ t.

Theorem 2.3.5. Gi s sp(M ) > 0. Let x ∈ m be a strict f -element of
Di−1 /Di for all i ≤ t. Then sp(M/xM ) ≤ sp(M ) − 1. The equality holds
if R is a quotient of local Cohen-Macaulay.


11

The equality sp(M/xM ) = sp(M ) − 1 in Theorem 2.3.4 may not
be valid if we drop the assumption that R is a quotient of local CohenMacaulay.
Example 2.3.6. For each integer r ≥ 0, there exists a Noetherian doamin
(R , m ) and a strict f -element a ∈ m of R∗ such that sp(R∗ ) = r + 1 and
sp(R∗ /aR∗ ) = −1.

2.4

A homological characterization of sequential polynomial type
Let Hm0 (M ) = Dt ⊂ ... ⊂ D0 = M be a dimension filtration of M and

di := dim Di for all i ≤ t. We stipulate dim Dt = −1 whenever Dt = 0.
Set Λ(M ) = {d0 , ..., dt }. Note that Λ(M ) \ {−1} = {dim(R/p) | p ∈
AssR M }. Suppose that R is a quotient of a Gorenstein local ring. Set q1 :=
max dim(K j (M )) and q2 := max p(K j (M )). The following theorem,

j ∈Λ(M
/
)

j∈Λ(M )

shows that sp(M ) can be computed in term of the deficiency modules

K j (M ).
Theorem 2.4.2. If R is a quotient of a Gorenstein local ring, then
sp(M ) = max{q1 , q2 }.
Corollary 2.4.3. Let r ≥ −1 be an integer. Suppose that R is a quotient
of a Gorenstein local ring. Then sp(M ) ≤ r if and only if dim K j (M ) ≤ r
for all j ∈
/ Λ(M ) and dim K j (M ) = j with p(K j (M )) ≤ r for all j ∈ Λ(M ).


12

CHAPTER 3
Index of reducibility of module
Through this chapter, let (R, m) be a Noether local ring and M a
finitely generated R-modulem with dim M = d, N be a submodule of M ,
A be an R-module Artinian. We denoted R and M the m-adic completion
of R and M respectively.

3.1

Index of reducibility of module Noetherian
Firstly, we recall the concept of index of redcibility of module. A

submodule N of M is called an irreducible submodule if N can not be written as an intersection of two properly larger submodules of M . Following
E. Noether, the number of irreducible components of an irredundant irreducible decomposition of N , which is independence of the choice of the
decompostion.
Definition 3.1.2. The number of irreducible components of an irredundant irreducible decomposition of N , which is independence of the choice
of the decompostion by E.Noether, is called the index of reducibility of N
and denoted by irM (N ). If q is a parameter ideal of M , then irM (qM ) is
said to be the index of reducibility of q on M .

We recall some results of J. D. Sally about the minimal number


13

of generators of module. Denote µ(N ) := dim R/m (N/mN ) the minimal
number of generators of N .
Set c(M ) = sup {µ(N ) | N is a submodule of M }. J. D. Sally proved that
c(R) < ∞ if and only if dim R ≤ 1. Then S. Goto v N. Suzuki improved
this result for modules. In a natural way, P.H. Qu defined the following
analogous notation for Artinian modules.
Notation 3.1.6. For an Artinian R-module A, set
r(A) = sup {dimR/m Soc(A/B) | B is a submodule ofA}.
For a finitely generated R-module N , set N ∗ = HomR (N, E(R/m)), the
Matlis duality of N . Then N ∗ is an Artinian R-module and c(N ) = r(N ∗ ).
For an Artinian R-module A, if R = R then A∗ = HomR (A, E(R/m)) is
a finitely generated R-module and r(A) = c(A∗ ). Note that r(A) < ∞ if
and only if dimR A ≤ 1.

3.2

Index of reducibility with the small sequential
polynomial type
P. H. Quy (2013) showed a uniform bound of irM (qM ) for all parame-

ter ideals q of M in the case where p(M ) ≤ 1. If p(M ) = 2, the question of
whether there exists a uniform bound of irM (qM ) for all parameter ideals
is still open. In case where p(M ) ≥ 3, a uniform bound of irM (qM ) for
all parameter ideals q of M may not exist even when sp(M ) = −1. In
fact, Goto and Suzuki (1984) constructed a sequentially Cohen-Macaulay

Noetherian local ring (R, m) such that p(R) = 3 and the supremum of
irR (q) is infinite, where q runs over all parameter ideals of R. On the other
hand, the notion of good parameter ideal (such ideals exist) makes an important role in the study of modules which are not necessarily unmixed.


14

Therefore, it is natural to ask if there exists a uniform bound of irM (qM )
for all good parameter ideals q of M . Some positive answers are given by
H. L. Truong (2013) for the case where sp(M ) = −1, and by P. H. Quy
(2012) for the case where sp(M ) ≤ 0. The answer for the case sp(M ) ≤ 1
will be solve in section 2 of this chapter. Before proving the main theorem,
we need some lemmas.
Lemma 3.2.1.Let sp(M ) ≤ 1 and R = R. Let H be a submodule of M
such that dim R H < d and p(M/H) ≤ 1. Let x ∈ m be a parameter of M
such that xH = 0. Then sp(M/xM ) ≤ 1.
For each Artinian R-module A with dim R (A) ≤ 1, the number r(A) is
defined as in Notation 3.1.8.
Lemma 3.2.2. Let R = R and µ = µ(m) the minimal number of generators of m. Let H be a submodule of M such that dim R H < d and
p(M/H) ≤ 1. Let x ∈ m be a parameter of M such that xH = 0. Then
dim R/m Soc(Hmd−1 (M/xM )) ≤ dim R/m Soc(Hmd (M )) + dim R/m Soc(Hmd−1 (H))
+ (µ + 1) r(Hmd−1 (M/H)) + µ r(Hmd−2 (M/H)).
From now on we use the following notations.
Notation 3.2.3. Let M = H0 ⊃ H1 ⊃ . . . ⊃ Hn be a filtration of
submodules of M satisfying the dimension condition. Let (x1 , . . . , xd ) be
a good parameter ideal of M with respect to this filtration. Let
M/xd M = H0 ⊃ H1 ⊃ . . . ⊃ Hm
be the filtration of submodules of M/xd M, where m and Hi are defined
as follows: If dim R H1 < d − 1, then we set m = n and Hi = (Hi +
xd M )/xd M ∼

= Hi ; If dim R H1 = d − 1, then we set m = n − 1 and Hi =
(Hi+1 + xd M )/xd M ∼
= Hi+1 , for i = 1, . . . , m.
Lemma 3.2.4. Let R = R. Let M = H0 ⊃ H1 ⊃ . . . ⊃ Hn be a filtration


15

of submodules of M satisfying the dimension condition such that p(Hn ) ≤ 1
and p(Hi−1 /Hi ) ≤ 1 for all i ≤ n. Let (x1 , . . . , xd ) be a good parameter
ideal with respect to this filtration. Then (x1 , . . . , xd−1 ) is a good parameter
ideal of M/xd M with respect to the filtration
M/xd M = H0 ⊃ H1 ⊃ . . . ⊃ Hm .
Moreover, p(Hm ) ≤ 1, dim R Hi < dim R Hi−1 and p(Hi−1 /Hi ) ≤ 1 for all
i = 1, . . . , m.

In the next lemma, let µ = µ(m) be the minimal number of generators
of m.
Lemma 3.2.5. Let sp(M ) ≤ 1. Let M = H0 ⊃ H1 ⊃ . . . ⊃ Hn ⊃ Hn+1 =
0 be a filtration of submodules of M satisfying the dimension condition.
Assume that p(Hi /Hi+1 ) ≤ 1 for all i ≤ n and Hn satisfies the following
condition: Hn = M when d ≤ 2, and dim R Hn ≥ 2 when d > 2. Set
hi = dim R Hi for i ≤ n. Then for any good parameter ideal q = (x1 , . . . , xd )
of M with respect to this filtration, we have
n

n

d

irM (qM ) ≤ µ

r

Hmj (Hi /Hi+1 )

dim R/m Soc(Hmhi (Hi )).

+

i=0 j
i=0

Now, we are ready to prove the main result of this section.
Theorem 3.2.6. Let sp(M ) ≤ 1 and Hm0 (M ) = Dt ⊂ ... ⊂ D0 = M be the
dimension filtration of M . Then for all good parameter ideals q of M with
respect to the filtration Dk ⊂ . . . ⊂ D1 ⊂ D0 = M with k ∈ {0, 1, . . . , t} be
the least integer such that p(Dk ) ≤ 1, we have
k−1

irM (qM ) ≤ µ

d

r Hmj (Di /Di+1 ) +
i=0 j
k

dim R/m Soc(Hmdi (Di )),

+
i=0

r Hmj (Dk )
j

16

where di := dim R Di for all i ≤ k and µ := µ(m) is the minimal number of
generators of m.
Remark 3.2.7. If q is a good parameter ideal with respect to the dimension filtration of M , then it is a good parameter ideal with respect to the
filtration M = D0 ⊃ D1 ⊃ . . . ⊃ Dk , where Di is the largest submodule
of M such that dim Di < dim Di−1 for all i and p(Dk ) ≤ 1. Moreover, if
p(M ) ≤ 1, then k = 0, in this case each parameter ideal of M is good with
respect to the filtration M = Dk . Therefore, Theorem 3.2.6 is an extension
of the main results of P.H. Quy (2013).

3.3

Index of reducibility of Artinian module and Duality Matlis
Throughout this section, we denoted E = E(R/m) is injective hull

of R/m, DR (−) = HomR (−, E) the Matlis dual functor and k = R/m is
residue field of R.
Following I. G. Macdonald (1973), A is said to be sum-irreducible if
A = 0 and A can not be expressed as the sum of two proper submodules of
A. Note that any Artinian R-module can be expressed as a sum of finitely

many sum-irreducible submodules, and the number of sum-irreducible submodules apprearing in such an irredundant sum-irreducible representation
does not depend on the choice of representation.
Definition 3.3.2. The number of sum-irreducible submodules apprearing
in an irredundant sum-irreducible representation of A is called index of
reducibility of A and denoted by irR (A).
It should be mentioned that in general, the reducibility index for
finitely generated modules is not preserved under m-adic completion. The
following lemma shows that the sum-reducibility index of an Artinian mod-


17

ule A is preserved under m-adic completion.
Lemma 3.3.3. irR (A) = irR A.
The following lemma give a formular to compute the index of reducibility of Artinian module is of finite length.
Lemma 3.3.7. Let M be a finitely generated R-module and A an Artinian
R-module. Set k = R/m. Then
(i) dimk Soc(M ) = dimk (DR (M )/m DR (M )).
(ii) If B be a submodule of A such that

R (A/B)

< ∞ and B + mA =

A then B = A.
(iii) If A is of finite length then irR (A) = dimk (A/mA).
Now, we are ready to prove next theorem, which is the main result
of this section, gives a comparison between the index of the submodule of
M with the index of Matlis duality of the corresponding quotient module
of M .

Theorem 3.3.10. Let R = R and N be a submodule of M . Then
irR (D(M/N )) ≤ irM (N ).
The equality holds true if and only if

R (M/N )

< ∞.


18

CONCLUSIONS OF THESIS
In this thesis, we have obtained the following results:
1. Introduce the notion of sequential polynomial type of M , which is
denote by sp(M ), in order to measure how far M is different from the
sequential Cohen-Macaulayness. We showed that sp(M) is dimension
of the non sequentially Cohen-Macaulay locus of M if R is a quotient
of Cohen-Macaulay local ring.
2. Study the change of the sequential polynomial type under localization, m-adic.
3. Show a relation between of sequential polynomial type of M and
M/xM for certain parameter x of M .
4. Describe sp(M ) in term of the deficiency modules of M when R is
a quotient of a Gorenstein local ring.
5. Give a formular uniform bound for index of reducibility of good
parameter q ca M vi sp(M ) ≤ 1;
6. Prove some basic properties for the index of reducibility of Artinian
module and show the relation between the index of the submodule of
M and the index reducibility of Matlis duality of a the corresponding
quotient module of M .

19

The results of the thesis were reported at the Seminars of Algebraic
and Number Theory Group, Thai Nguyen University and at many conferences such as Algebraic Conference - Geometry - Topology, Buon Ma
Thuot (October 2016), Vietnam-Japan, Thai Nguyen Association Conference (1/2017), 9th National Mathematical Conference, Nha Trang (8/2018),
Vietnam-Japan Commutative Algebra Conference, Hue, (9 / 2018).

Papers used in the thesis
[1] L.T. Nhan, T.D. Dung and T.D.M. Chau, ”A measure of nonsequential Cohen-Macaulayness of finitely generated modules”, J. Algebra,
468 (2016), 275-295.
[2] T.D. Dung and L.T. Nhan, ”A uniform bound of reducibility
index of good parameter ideals for certain class of modules”, J. Pure Appl.
Algebra, 223 (2019), 3964-3979.
[3] T.D. Dung, ”On the invariant of the index of reducibility for
parameter ideals of Cohen-Macaulay modules”, Thai Nguyen University
Journal of Science and Technology, 147(02) (2015), 199202.
[4] N.T. Cuong, T.D. Dung and L.T. Nhan, ”Reducibility index of
finitely generated modules and Matlis duality”, (2019), Preprint.