Co-intersection graph of submodules of a module
Let M be a unitary left R-module where R is a ring with identity. The co-intersection graph of proper submodules of M, denoted by Ω(M), is an undirected simple graph whose the vertex set V(Ω) is a set of all non-trivial submodules of M and there is an edge between two distinct vertices N and K if an...
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irk-123456789-1551962019-06-17T01:26:09Z Co-intersection graph of submodules of a module Mahdavi, L.A. Talebi, Y. Let M be a unitary left R-module where R is a ring with identity. The co-intersection graph of proper submodules of M, denoted by Ω(M), is an undirected simple graph whose the vertex set V(Ω) is a set of all non-trivial submodules of M and there is an edge between two distinct vertices N and K if and only if N+K≠M. In this paper we investigate connections between the graph-theoretic properties of Ω(M) and some algebraic properties of modules . We characterize all of modules for which the co-intersection graph of submodules is connected. Also the diameter and the girth of Ω(M) are determined. We study the clique number and the chromatic number of Ω(M). 2016 Article Co-intersection graph of submodules of a module / L.A. Mahdavi, Y. Talebi // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 128-143. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC:05C15, 05C25, 05C69, 16D10. http://dspace.nbuv.gov.ua/handle/123456789/155196 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let M be a unitary left R-module where R is a ring with identity. The co-intersection graph of proper submodules of M, denoted by Ω(M), is an undirected simple graph whose the vertex set V(Ω) is a set of all non-trivial submodules of M and there is an edge between two distinct vertices N and K if and only if N+K≠M. In this paper we investigate connections between the graph-theoretic properties of Ω(M) and some algebraic properties of modules . We characterize all of modules for which the co-intersection graph of submodules is connected. Also the diameter and the girth of Ω(M) are determined. We study the clique number and the chromatic number of Ω(M). |
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Article |
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Mahdavi, L.A. Talebi, Y. |
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Mahdavi, L.A. Talebi, Y. Co-intersection graph of submodules of a module Algebra and Discrete Mathematics |
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Mahdavi, L.A. Talebi, Y. |
| author_sort |
Mahdavi, L.A. |
| title |
Co-intersection graph of submodules of a module |
| title_short |
Co-intersection graph of submodules of a module |
| title_full |
Co-intersection graph of submodules of a module |
| title_fullStr |
Co-intersection graph of submodules of a module |
| title_full_unstemmed |
Co-intersection graph of submodules of a module |
| title_sort |
co-intersection graph of submodules of a module |
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Інститут прикладної математики і механіки НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/155196 |
| citation_txt |
Co-intersection graph of submodules of a module / L.A. Mahdavi, Y. Talebi // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 128-143. — Бібліогр.: 11 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT mahdavila cointersectiongraphofsubmodulesofamodule AT talebiy cointersectiongraphofsubmodulesofamodule |
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2025-07-14T07:16:21Z |
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2025-07-14T07:16:21Z |
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1837605722673643520 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 1, pp. 128–143
© Journal “Algebra and Discrete Mathematics”
Co-intersection graph
of submodules of a module
Lotf Ali Mahdavi and Yahya Talebi
Communicated by D. Simson
Abstract. Let M be a unitary left R-module where R is a
ring with identity. The co-intersection graph of proper submodules
of M , denoted by Ω(M), is an undirected simple graph whose the
vertex set V (Ω) is a set of all non-trivial submodules of M and
there is an edge between two distinct vertices N and K if and
only if N + K 6= M . In this paper we investigate connections
between the graph-theoretic properties of Ω(M) and some algebraic
properties of modules . We characterize all of modules for which the
co-intersection graph of submodules is connected. Also the diameter
and the girth of Ω(M) are determined. We study the clique number
and the chromatic number of Ω(M).
1. Introduction
The investigation of the interplay between the algebraic structures-
theoretic properties and the graph-theoretic properties has been studied
by several authors. As a pioneer, J. Bosak [4] in 1964 defined the graph of
semigroups. Inspired by his work, B. Csakany and G. Pollak [7] in 1969,
studied the graph of subgroups of a finite group. The Intersection graphs
of finite abelian groups studied by B. Zelinka [11] in 1975. Recently,
in 2009, the intersection graph of ideals of a ring, was considered by
I. Chakrabarty et. al. in [5]. In 2012, on a graph of ideals researched
2010 MSC: 05C15, 05C25, 05C69, 16D10.
Key words and phrases: co-intersection graph, clique number, chromatic num-
ber.
L. A. Mahdavi, Y. Talebi 129
by A. Amini et. al. in [2] and Also, intersection graph of submodules of
a module introduced by S. Akbari et. al. in [1]. Motivated by previous
studies on the intersection graph of algebraic structures, in this paper we
define the co-intersection graph of submodules of a module. Our main
goal is to study the connection between the algebraic properties of a
module and the graph theoretic properties of the graph associated to it.
Throughout this paper R is a ring with identity and M is a unitary
left R-module. We mean from a non-trivial submodule of M is a nonzero
proper left submodule of M .
The co-intersection graph of an R-module M , denoted by Ω(M), is
defined the undirected simple graph with the vertices set V (Ω) whose
vertices are in one to one correspondence with all non-trivial submodules
of M and two distinct vertices are adjacent if and only if the sum of the
corresponding submodules of M is not-equal M .
A submodule N of an R-module M is called superfluous orsmall
in M (we write N ≪ M), if for every submodule X ⊆ M , the equality
N + X = M implies X = M , i.e., a submodule N of M is called small
in M , if N + L 6= M for every proper submodule L of M . The radical of
R-module M written Rad(M), is sum of all small submodules of M .
A non-zero R-module M is called hollow, if every proper submodule
of M is small in M .
A non-zero R-module M is called local, if has a largest submodule,
i.e., a proper submodule which contains all other proper submodules.
An R-module M is said to be A-projective if for every epimorphism
g : A → B and homomorphism f : M → B, there exists a homomorphism
h : M → A, such that gh = f . A module P is projective if P is A-
projective for every R-module A. If P is P -projective, then P is also
called self-(or quasi-)projective.
A non-zero R-module M is said to be simple, if it has no non-trivial
submodule. A nonzero R-module M is called indecomposable, if it is not a
direct sum of two non-zero submodules. For an R-module M , the length
of M is the length of composition series of M , denoted by lR(M).
An R-module M has finite length if lR(M) < ∞, i.e., M is Noetherian
and Artinian. The ring of all endomorphisms of an R-module M is denoted
by EndR(M).
Let Ω=(V (Ω),E(Ω)) be a graph with vertex set V (Ω) and edge set E(Ω)
where an edge is an unordered pair of distinct vertices of Ω. Graph Ω
is finite, if Card(V (Ω)) < ∞, otherwise Ω is infinite. A subgraph of a
graph Ω is a graph Γ such that V (Γ) ⊆ V (Ω) and E(Γ) ⊆ E(Ω). By order
130 Co-intersection graph of submodules of a module
of Ω, we mean the number of vertices of Ω and we denoted it by |Ω|. If
X and Y are two adjacent vertices of Ω, then we write X ↔ Y .
The degree of a vertex υ in a graph Ω, denoted by deg(υ), is the number
of edges incident with υ. A vertex v is called isolated if deg(v) = 0. Let
U and V be two distinct vertices of Ω. An U , V -path is a path with
starting vertex U and ending vertex V . For distinct vertices U and V ,
d(U, V ) is the least length of an U , V -path. If Ω has no such a path, then
d(U, V ) = ∞. The diameter of Ω, denoted by diam(Ω) is the supremum
of the set {d(U, V ): U and V are distinct vertices of Ω}.
A cycle in a graph is a path of length at least 3 through distinct
vertices which begins and ends at the same vertex. We mean of (X, Y, Z)
is a cycle of length 3. The girth of a graph is the length of its shortest
cycle. A graph with no-cycle has infinite girth.
By a null graph, we mean a graph with no edges. A graph is said to be
connected if there is a path between every pair of vertices of the graph.
A tree is a connected graph which does not contain a cycle.
A star graph is a tree consisting of one vertex adjacent to all the
others.
A complete graph is a graph in which every pair of distinct vertices are
adjacent. The complete graph with n distinct vertices, denoted by Kn.
By a clique in a graph Ω, we mean a complete subgraph of Ω and the
number of vertices in a largest clique of Ω, is called the clique number
of Ω and is denoted by ω(Ω).
An independent set in a graph is a set of pairwise non-adjacent vertices.
An independence number of Ω, written α(Ω), is the maximum size of an
independent set.
For a graph Ω, let χ(Ω), denote the chromatic number of Ω, i.e., the
minimum number of colors which can be assigned to the vertices of Ω
such that every two adjacent vertices have different colors.
2. Connectivity, diameter and girth of Ω(M)
In this section, we characterize all modules for which the co-intersection
graph of submodules is not connected. Also the diameter and the girth
of Ω(M) are determined. Finally we study some modules whose co-
intersection graphs are complete.
Theorem 2.1. Let M be an R-module. Then the graph Ω(M) is not
connected if and only if M is a direct sum of two simple R-modules.
L. A. Mahdavi, Y. Talebi 131
Proof. Assume that Ω(M) is not connected. Suppose that Ω1 and Ω2 are
two components of Ω(M). Let X and Y be two submodules of M such
that X ∈ Ω1 and Y ∈ Ω2. Since there is no X,Y -path, then M = X + Y .
Now, if X ∩ Y 6= (0), then by
X ∩ Y + X = X 6= M and X ∩ Y + Y = Y 6= M
implies that there is a X,Y -path by X ∩ Y , to form X ↔ X ∩ Y ↔ Y ,
a contradiction. Hence, X ∩ Y = (0) and M = X ⊕ Y . Now, we show
that X and Y are minimal submodules of M . To see this, let Z be a
submodule of M such that (0) 6= Z ⊆ X then Z + X = X 6= M . Hence
Z and X are adjacent vertices, which implies that Z ∈ Ω1. Hence there
is no Z,Y -path and by arguing as above, we have M = Z + Y , since Z
and Y are not adjacent vertices. But since
X = X ∩ M = X ∩ (Z + Y ) = Z + X ∩ Y = Z
by Modularity condition, X is a minimal submodule of M .
A similar argument shows that Y is also a minimal submodule of M
and in fact every non-trivial submodule of M is a minimal submodule,
which yields that every non-trivial submodule is also maximal. But,
minimality of X and Y implies that, they are simple R-modules and since
M = X ⊕ Y , we are done.
Conversely, suppose that Ω(M) is connected. Let M = X ⊕ Y , where
X and Y are simple R-modules. Let M1 = X × {0} and M2 = {0} × Y .
Then M1 and M2 are minimal submodules of M . Moreover, M1 and
M2 are simple R-modules. But, M = M1 ⊕ M2 and M1
∼= M/M2 and
M2
∼= M/M1. Consequently, M1 and M2 are maximal submodules of M .
Therefore, M1 and M2 are two maximal and minimal submodules of M .
We show that M1 is an isolated vertex in Ω(M). To see this, let N be
a vertex in Ω(M), with N + M1 6= M . Then, maximality of M1 implies
that N + M1 = M1, and hence N ⊆ M1. Then, minimality of M1 implies
that M1 = N . Hence, M1 is an isolated vertex in Ω(M). Thus, Ω(M) is
not connected, a contradiction. This completes the proof.
Example 2.2. Let Zpq be a Z-module, such that p and q are two distinct
prime numbers. Then Ω(Zpq) is not connected. Because, Zp and Zq are
simple Z-modules and by Theorem 2.1, Ω(Zp ⊕ Zq) is not connected.
Since Zpq
∼= Zp ⊕Zq, Ω(Zpq) is not connected. But, we consider Zp1p2p3
as
Z-module, such that pi is a prime number, for i = 1, 2, 3. We know M1 =
p1Zp1p2p3
, M2 = p2Zp1p2p3
and M3 = p3Zp1p2p3
are the only maximal
132 Co-intersection graph of submodules of a module
submodules of Zp1p2p3
. Also, K = p1p2Zp1p2p3
, L = p1p3Zp1p2p3
and
N = p2p3Zp1p2p3
are the other submodules of Zp1p2p3
. Hence, Ω(Zp1p2p3
)
is connected (see Fig. 1).
M2 N M3
K L
M1
Figure 1. Ω(Zp1p2p3
).
Corollary 2.3. Let M be an R-module. If Ω(M) is connected, then the
following hold:
(1) every pair of maximal submodules of M , have non-trivial intersection,
and there exists a path between them;
(2) every pair of minimal submodules of M , have non-trivial sum, and
there is an edge between them.
Proof. (1) Let M1 and M2 be two maximal submodules of M . Clearly,
M1 ∩ M2 6= M . Let M1 ∩ M2 = (0). Since M = M1 + M2, M = M1 ⊕ M2.
So M/M1
∼= M2 and M/M2
∼= M1, hence M1 and M2 are two simple
R-modules. Now, by Theorem 2.1, Ω(M) is not connected, which is a
contradiction by hypothesis. Hence M1 ∩ M2 6= (0), and there exists a
path to form M1 ↔ M1 ∩ M2 ↔ M2 between them.
(2) Let M1 and M2 be two minimal submodules of M such that
M = M1 + M2. If M1 ∩ M2 = (0), then M = M1 ⊕ M2, such that M1
and M2 are two simple R-modules, then by Theorem 2.1, Ω(M) is not
connected, which is a contradiction by hypothesis. Also if M1 ∩ M2 6= (0),
since (0) $ M1 ∩ M2 ⊆ Mi $ M , for i = 1, 2, by minimality of M1
and M2 implies that M1 ∩ M2 = M1 = M2, which is a contradiction by
hypothesis M1 6= M2. Therefore, M 6= M1 + M2, and there is an edge
between them.
L. A. Mahdavi, Y. Talebi 133
Corollary 2.4. Let M be an R-module. If |Ω(M)| > 2, and Ω(M) is not
connected, then the following hold:
(1) Ω(M) is a null graph;
(2) lR(M) = 2.
Proof. (1) Suppose that Ω(M) is not connected, then by Theorem 2.1,
M = M1 ⊕ M2, such that M1 and M2 are two simple R-modules. So any
non-trivial submodule of M is simple. In fact any non-trivial submodule
of M is minimal and consequently a maximal submodule. Hence for each
two distinct non-trivial submodules K and L of M , we have M = K + L,
thus there is no edge between two distinct vertices K and L of the graph
Ω(M). Therefore, Ω(M) is a null graph.
(2) It is clear by Theorem 2.1.
Theorem 2.5. Let M be an R-module. If Ω(M) is connected, then
diam(Ω(M)) 6 3.
Proof. Let A and B be two non-trivial distinct submodules of M . If
A + B 6= M then A and B are adjacent vertices of Ω(M), so d(A, B) = 1.
Suppose that A + B = M . If A ∩ B 6= (0), then there exists a path
A ↔ A ∩ B ↔ B of length 2, so d(A, B) = 2. Now, if A ∩ B = (0), then
M = A ⊕ B, and since Ω(M) is connected, by Corollary 2.3(1), implies
that at least one of A and B should be non-maximal. Assume that B is not
maximal. Hence there exists a submodule X of M such that B $ X $ M ,
and B + X = X 6= M . Now, if A + X 6= M , then there exists a path
A ↔ X ↔ B of length 2, then d(A, B) = 2. But, if A + X = M , then
by Modularity condition, X = X ∩ (A ⊕ B) = (X ∩ A) ⊕ B. Now, if
X ∩ A = (0), then X = B, a contradiction with existence X. Also, if
X ∩ A 6= (0), then there exists a path A ↔ X ∩ A ↔ X ↔ B of length 3,
so d(A, B) 6 3. Therefore, diam(Ω(M)) 6 3.
Remark 2.6. Let R be an integral domain. Then Ω(R) is a connected
graph with diam(Ω(R)) = 2.
Proof. Suppose that I and J are two ideals of integral domain R. Now,
if I + J 6= R, then I and J are adjacent vertices, then d(I, J) = 1. But,
if I + J = R, there exist two possible cases I ∩ J = (0) or I ∩ J 6= (0).
The first case implies that R = I ⊕ J , then there is idempotent e in R,
such that I = Re and J = R(1 − e). Since integral domain R has no
zero divisor, then e = 0 or e = 1, thus I = (0) and J = R or I = R and
J = (0), this is a contradiction. In second case, since IJ = I ∩ J 6= (0)
134 Co-intersection graph of submodules of a module
and I + IJ = I 6= R, J + IJ = J 6= R, then there exists a path to form
I ↔ IJ ↔ J , then d(I, J) = 2. Consequently, Ω(R) is a connected graph
and diam(Ω(R)) = 2.
Theorem 2.7. Let M be an R-module, and Ω(M) a graph, which contains
a cycle. Then girth(Ω(M)) = 3.
Proof. On the contrary, assume that girth(Ω(M)) > 4. This implies
that every pair of distinct non-trivial submodules M1 and M2 of M with
M1+M2 6= M should be comparable. Because, if X and Y are two distinct
non-trivial submodules of M with X + Y 6= M such that X *Y and
Y * X, then X $ X+Y and Y $ X+Y . As X+Y +X = X+Y 6= M and
Y + X + Y = X + Y 6= M , hence Ω(M) has a cycle to form (X, X + Y, Y )
of length 3, a contradiction. Now, since girth(Ω(M)) > 4, Ω(M) contains
a path of length 3, say A ↔ B ↔ C ↔ D. Since every two submodules
in this path are comparable and every chain of non-trivial submodules
of length 2 induces a cycle of length 3 in Ω(M), the only two possible
cases are A ⊆ B, C ⊆ B or B ⊆ A, B ⊆ C, D ⊆ C. The first case
yields A + B = B 6= M , C + B = B 6= M , A + C ⊆ B 6= M , then
(A, B, C) is a cycle of length 3 in Ω(M), a contradiction. In the second
case, we have B + A = A 6= M , B + C = C 6= M , B + D ⊆ C 6= M and
C + D = C 6= M , then (B, C, D) is a cycle of length 3 in Ω(M), which
again this is a contradiction. Consequently, girth(Ω(M)) = 3, and the
proof is complete.
Example 2.8. Since Z is an integral domain, then by Remark 2.6, Ω(Z) is
a connected graph and contains a cycle (2Z, 4Z, 6Z), then by Theorem 2.7,
girth(Ω(Z)) = 3.
Theorem 2.9. Let M be a Noetherian R-module. Then, Ω(M) is complete
if and only if M contains a unique maximal submodule.
Proof. Suppose that M is a Noetherian R-module, then M has at least one
maximal submodule. Moreover every nonzero submodule of M contained
in a maximal submodule. Therefore, if M possesses a unique maximal
submodule, say U , then U contains every nonzero submodule of M . As-
sume that K and L are two distinct vertices of Ω(M). Then K ⊆ U and
L ⊆ U , hence K + L ⊆ U 6= M . Therefore, Ω(M) is complete.
Conversely, suppose that Ω(M) is complete. Let X and Y be two
distinct maximal submodules of M . Then X + Y 6= M , since X ⊆ X + Y
and Y ⊆ X + Y , by maximality of X and Y , we have X + Y = X = Y , a
L. A. Mahdavi, Y. Talebi 135
contradiction. Consequently, M contains a unique maximal submodule,
and the proof is complete.
Theorem 2.10. Let M be an Artinian R-module. Then Ω(M) is con-
nected if and only if M contains a unique minimal submodule.
Proof. Suppose that M is an Artinian R-module, then M has at least
one minimal submodule. Moreover, every nonzero submodule of M con-
tains a minimal submodule. Therefore, if M possesses a unique minimal
submodule, say L, then L contained in every nonzero submodule of M .
Assume that A and B are two distinct vertices of Ω(M). Then L ⊆ A
and L ⊆ B, hence L + A = A 6= M and L + B = B 6= M . Then there is
A, B-path, to form A ↔ L ↔ B. Therefore, Ω(M) is connected.
Conversely, suppose that Ω(M) is connected. Let N1 and N2 be
two distinct minimal submodules of M . Since (0) ⊆ N1 ∩ N2 ⊆ Ni $
M , for i = 1, 2, by minimality of N1 and N2, if N1 ∩ N2 6= (0), then
N1 ∩ N2 = N1 = N2, a contradiction. If N1 ∩ N2 = (0), then the only
two possible cases are N1 + N2 = M or N1 + N2 6= M . If N1 + N2 = M ,
then M = N1 ⊕ N2 such that N1 and N2 are two simple R-modules.
Then by Theorem 2.1, Ω(M) is not connected, a contradiction. But, if
N1 +N2 6= M , N1 = N1/(N1 ∩N2) ∼= (N1 +N2)/N2 and N1 is simple, then
N2 is maximal submodule of M . Also, similarly N1 is maximal submodule
of M . Since, (0) $ Ni ⊆ N1 + N2 $ M , for i = 1, 2, by maximality of N1
and N2, we have N1 + N2 = N1 = N2, which again this is a contradiction.
Consequently, M contains a unique minimal submodule, and the proof is
complete.
Proposition 2.11. Let M be an R-module, with the graph Ω(M). Then
M is a hollow if and only if Ω(M) is a complete graph.
Proof. Suppose that K1 and K2 are two distinct vertices of Ω(M). Since
M is a hollow R-module, then K1 ≪ M and K2 ≪ M . Then by [3,
Proposition 5.17(2)] K1 + K2 ≪ M . Thus, K1 + K2 6= M . Therefore,
Ω(M) is a complete graph.
Conversely, assume that Ω(M) is a complete graph. Let N is a non-
trivial submodule of M . Since Ω(M) is complete, N is adjacent to every
other vertex of Ω(M). Then N + X 6= M , for every proper submodule X
of M , thus N ≪ M . Hence, M is a hollow R-module.
Corollary 2.12. Let M be an R-module and N be a non-trivial submodule
of M . If |Ω(M)| = n, then N is a non-trivial small submodule of M if
and only if deg(N) = n − 1, n ∈ N.
136 Co-intersection graph of submodules of a module
Proof. It is clear.
Example 2.13. We consider Z12 as Z12- module. The non-trivial sub-
modules of Z12 are M1 = {0, 6}, M2 = {0, 4, 8}, M3 = {0, 3, 6, 9},
M4 = {0, 2, 4, 6, 8, 10} such that M1 = {0, 6} is the only non-trivial small
submodule of Z12 and |Ω(Z12)| = 4. Then, by Corollary 2.12, deg(M1) = 3
(see Fig. 2).
M1 M2
M3M4
Figure 2. Ω(Z12).
Example 2.14. For every prime number p and for all n ∈ N with n > 2,
the co-intersection graph of Z-module Zpn , is a complete graph. Because,
Z-module Zpn is local, then it is hollow. Hence, by Proposition 2.11,
Ω(Zpn) is complete. Also, since the number of non-trivial submodules of
Z-module Zpn is equal n − 1. Therefore, Ω(Zpn) is a complete graph with
n − 1 vertices, i.e., Ω(Zpn)=Kn−1 (see Fig. 3 for p = 2 and n = 5).
M1 M2
M3M4
Figure 3. Ω(Z32).
Example 2.15. For every prime number p, the co-intersection graph of
Z-module Zp∞ , is a complete graph. Because, by [10, 41.23, Exercise (6)],
for every prime number p, the Z-module Zp∞ is hollow. Therefore, by
Proposition 2.11, Ω(Zp∞) is complete.
L. A. Mahdavi, Y. Talebi 137
Corollary 2.16. Let M be an R-module. Then Ω(M) is complete, if one
of the following holds:
(1) if M is an indecomposable R-module, such that every pair of non-
trivial submodules of M , have zero intersection;
(2) if M is a local R-module;
(3) if M is a self-(or quasi-) projective R-module and EndR(M) is a
local ring.
Proof. (1) It is clear by definition.
(2) Since local R-modules are hollow, it follows from Proposition 2.11.
(3) Since, M is a self- (or quasi-) projective R-module and EndR(M)
is a local ring, M is hollow by [9, Proposition 2.6]. Then it follows from
Proposition 2.11.
3. Clique number, chromatic number and some finiteness
conditions
Let M be an R-module. In this section, we obtain some results on the
clique and the chromatic number of Ω(M). We also study the condition
under which the chromatic number of Ω(M) is finite. Finally, it is proved
that χ(Ω(M)) is finite, provided ω(Ω(M)) is finite.
Lemma 3.1. Let M be an R-module and ω(Ω(M)) < ∞. Then the
following hold:
(1) lR(M) < ∞;
(2) ω(Ω(M)) = 1 if and only if either |Ω(M)| = 1 or |Ω(M)| > 2 and M
is a direct sum of two simple R-modules(i.e., Ω(M) is null);
(3) if ω(Ω(M)) > 1, then the number of minimal submodules of M is
finite.
Proof. (1) Let M0 ⊂ M1 ⊂ · · · ⊂ Mi ⊂ Mi+1 ⊂ . . . , be an infinite strictly
increasing sequence of submodules of M . For i < j, Mi + Mj = Mj 6= M ,
so similarly for infinite strictly decreasing sequence of submodules of M .
Hence, any infinite strictly increasing or decreasing sequence of submodules
of M induces a clique in Ω(M) which contradicts the finiteness ω(Ω).
This implies that for infinite strictly (increasing and decreasing) sequence
of submodules of M , Mn = Mn+i for i = 1, 2, 3, . . . . Thus, M should be
Noetherian and Artinian. Therefore, lR(M) < ∞.
(2) Suppose that ω(Ω) = 1 and |Ω(M)| > 2. This implies that Ω(M)
is not connected. Hence, by Theorem 2.1, M is a direct sum of two simple
R-modules.
138 Co-intersection graph of submodules of a module
Conversely, it is clear by Theorem 2.1.
(3) Since ω(Ω) > 1, by Part (2), M is not a direct sum of two simple
R-modules. Then, by Theorem 2.1, Ω(M) is not connected. Therefore, by
Corollary 2.3(2), every pair of minimal submodules of M , have non-trivial
sum. Suppose that Ω⋆(M) is a subgraph of Ω(M) with the vertex set
V ⋆={L 6 M |L is minimal submodule ofM}. Then Ω⋆(M) is a clique
in M , and Card(V ⋆) = ω(Ω⋆(M)) 6 ω(Ω(M)) < ∞. Hence, then the
number of minimal submodules of M is finite.
Remark 3.2. Let M be an R-module with the length lR(M) and N be
a submodule of M and △(Ω) = max{deg(vi)|vi ∈ V (Ω)}, then:
(1) Clearly, ω(Ω(N)) 6 ω(Ω(M)) and ω(Ω(M/N)) 6 ω(Ω(M)). Hence,
ω(Ω(M)) < ∞, implies that ω(Ω(N)) < ∞ and ω(Ω(M/N)) < ∞.
(2) Clearly, lR(M) 6 ω(Ω(M))+1. Also if Ω(M) is a connected graph,
then ω(Ω(M)) 6 χ(Ω(M)) 6 △(Ω) + 1 by Theorem 10.3(1) of [6, p. 289].
Hence, △(Ω) < ∞, implies that χ(Ω(M)) < ∞, ω(Ω(M)) < ∞, and
lR(M) < ∞.
Theorem 3.3. Let M be an R-module and |Ω(M)| > 2. Then the fol-
lowing conditions are equivalent:
(1) Ω(M) is a star graph;
(2) Ω(M) is a tree;
(3) χ(Ω(M) = 2;
(4) lR(M) = 3, M has a unique minimal submodule L such that every
non-trivial submodule contains L is maximal submodule of M .
Proof. (1) ⇒ (2) ⇒ (3) It follows from definitions.
(3) ⇒ (4), Let χ(Ω(M) = 2. Then Ω(M) is not null and by Corol-
lary 2.4(1), Ω(M) is connected. By Remark 3.2(2), ω(Ω(M)) 6 χ(Ω(M)),
hence ω(Ω(M)) < ∞ and by Lemma 3.1(1), lR(M) < ∞. Then M is
Artinian. Hence M contains a minimal submodule L. We show that L is
unique. Let there exist two minimal submodules L1 and L2 of M . Then
by Corollary 2.3(2), L1 + L2 6= M . Since (L1 + L2) + L1 = L1 + L2 6= M
and (L1 + L2) + L2 = L1 + L2 6= M , then (L1, L1 + L2, L2) is a cycle of
length 3 in Ω(M), which contradicts χ(Ω(M) = 2. Hence, L is a unique
minimal submodule of M . Suppose that L contained in every non-trivial
submodule of M . If K is a non-trivial submodule of M such that L $ K,
we show that K is a maximal submodule of M . Let L $ K $ X $ M ,
since L + K = K, K + X = X and L + X = X, (L, K, X) is a cycle of
L. A. Mahdavi, Y. Talebi 139
length 3, which is a contradiction. Consequently, K is a maximal submod-
ule contains L, and (0) $ L $ K $ M , is a composition series of M with
length 3. Therefore, lR(M) = 3.
(4) ⇒ (1) Suppose that lR(M) = 3 and M has a unique minimal
submodule L, such that every non-trivial submodule Li, (i ∈ I) of M
contains L, is a maximal submodule of M . Then, (0) $ L $ Li $ M , for all
i ∈ I, are composition series of M with length 3, such that Li+L = Li 6= M
and Li + Lj = M for i 6= j, Therefore, Ω(M) is a star graph. The proof
is complete.
Lemma 3.4. Let M be an R-module and N a vertex of the graph Ω(M).
If deg(N) < ∞, then lR(M) < ∞.
Proof. Suppose that N contains an infinite strictly increasing sequence
of submodules N0 ⊂ N1 ⊂ N2 ⊂ · · · . Then Ni + N = N 6= M , for
all i ∈ I, which contradicts deg(N) < ∞. Similarly, if N contains an
infinite strictly decreasing sequence of submodules, which again yields
a contradiction. Also assume that M/N contains an infinite strictly
increasing sequence of submodules M0/N ⊂ M1/N ⊂ M2/N ⊂ · · · . Since
N ⊂ M0 ⊂ M1 ⊂ M2 ⊂ · · · . Then Mi + N = Mi 6= M , for all i ∈ I, a
contradiction. Similarly, if M/N contains an infinite strictly decreasing
sequence of submodules, which again yields a contradiction. Hence, N
and M/N can not contain an infinite strictly increasing or decreasing
sequence of submodules. Thus, they are Noetherian R-module as well as
Artinian R-module. Hence, M is Noetherian R-module as well as Artinian
R-module. Therefore, lR(M) < ∞.
Lemma 3.5. Let M be an R-module and N a minimal submodule of M .
Assume that L is a non-trivial submodule of M such that L + N = M .
Then, L is a maximal submodule of M .
Proof. Let U be a submodule of M such that (0) 6= L ⊆ U $ M .
Then L + U = U and (0) ⊆ L ∩ N ⊆ U ∩ N ⊆ N . Since N is a
minimal submodule of M , L ∩ N = N or U ∩ N = (0). If L ∩ N = N
then N ⊆ L thus N + U ⊆ L + U = U and M = L + N ⊆ N + U ,
implies that M = U , which is a contradiction. Hence U ∩ N = (0) and
U = U ∩ (L + N) = L + U ∩ N = L by Modularity condition. Therefore,
L is a maximal submodule of M .
Theorem 3.6. Let M be an R-module with the graph Ω(M) and N is a
minimal submodule of M , such that deg(N) < ∞. If Ω(M) is connected,
then the following hold:
140 Co-intersection graph of submodules of a module
(1) the number of minimal submodules of M is finite;
(2) χ(Ω(M)) < ∞;
(3) if Rad(M) 6= (0), then Ω(M) has a vertex which is adjacent to every
other vertex.
Proof. (1) Let Σ = {K 6 M | K be a minimal submodule ofM}. Clearly,
Σ 6= ∅. Since Ω(M) is connected, then by Corollary 2.3(2), for all K ∈ Σ,
K + N 6= M , for N and every K ∈ Σ are minimal submodules of M and
adjacent vertices of Ω(M) with deg(N) < ∞. Hence, Card(Σ) < ∞, thus
the number of minimal submodules of M is finite.
(2) Let {Ui}i∈I be the family of non-trivial submodules which are not
adjacent to N . Thus, Ui + N = M , for all i ∈ I. Hence by Lemma 3.5, Ui
is maximal submodule of M , for all i ∈ I. Since Ui + Uj = M , for i 6= j,
distinct vertices Ui and Uj are not two adjacent vertices of Ω(M). Hence,
one can color all {Ui}i∈I by a color, and other vertices, which are a finite
number of adjacent vertices N , by a new color to obtain a proper vertex
coloring of Ω(M). Therefore, χ(Ω(M)) < ∞.
(3) In order to establish this part, consider Rad(M). Since N is a
vertex of Ω(M) and deg(N) < ∞, by Lemma 3.4, lR(M) < ∞ and thus M
is Noetherian. Then by [3, Proposition 10.9], M is finitely generated and by
[3, Theorem 10.4(1)], Rad(M) ≪ M . Now, we know that Rad(M) 6= M ,
otherwise, M = (0), which is a contradiction. Hence, Rad(M) is a non-
trivial submodule of M and for each non-trivial submodule K of M , we
have K + Rad(M) 6= M . Consequently, Ω(M) has the vertex Rad(M),
which is adjacent to every other vertex.
Corollary 3.7. Let M be an R-module with the graph Ω(M). Then the
following hold:
(1) if M has no maximal or no minimal submodule, then Ω(M) is
infinite;
(2) if M contains a minimal submodule and every minimal submodule
of M has finite degree, then Ω(M) is either null or finite.
Proof. (1) If M has no maximal submodule, since (0) $ M and (0) is not
maximal, there exists a submodule M0 of M , such that (0) $ M0 $ M ,
and M0 is not maximal, there exists a submodule M1 of M , such that
(0) $ M0 $ M1 $ M . Consequently, there exists (0) $ M1 $ M1 $ · · · $
M , and for i < j, Mi + Mj = Mj 6= M . Thus M contains an infinite
strictly increasing sequence of submodules. Therefore, Ω(M) is infinite.
If M has no minimal submodule, since M % (0) and M is not minimal,
L. A. Mahdavi, Y. Talebi 141
there exists a submodule N0, such that M % N0 % (0), and N0 is not
minimal, there exists a submodule N1, such that M % N0 % N1 % (0).
Consequently, there exists M % N0 % N1 % · · · % (0), and for i < j,
Ni + Nj = Ni 6= M . Thus M contains an infinite strictly decreasing
sequence of submodules. Therefore, Ω(M) is infinite.
(2) Suppose that Ω(M) is not null and by contrary assume that
Ω(M) is infinite. Since Ω(M) is not null, by Corollary 2.4(1), Ω(M) is
connected and since every minimal submodule of M has finite degree, by
Lemma 3.4, lR(M) < ∞. Hence, M is Artinian and by Theorem 3.6(1),
the number of minimal submodules of M is finite. Since Ω(M) is infinite,
and V (Ω(M)) = {Ni|i ∈ I}, there exists a minimal submodule N which
N ⊆ Ni, for each i ∈ I, then N + Ni = Ni 6= M , for each i ∈ I. This
contradicts deg(N) < ∞. Hence, Ω(M) is a finite graph.
Theorem 3.8. Let M be an R-module such that Ω(M) is infinite and
ω(Ω(M)) < ∞. Then the following hold:
(1) the number of maximal submodules of M is infinite;
(2) the number of non-maximal submodules of M is finite;
(3) χ(Ω(M)) < ∞;
(4) α(Ω(M)) = ∞.
Proof. (1) On the contrary, assume that the number of maximal submod-
ules of M is finite. Since Ω(M) is infinite, Ω(M) has an infinite clique
which contradicts the finiteness of ω(Ω(M)).
(2) Suppose that ω(Ω(M)) < ∞, then by Lemma 3.1(1), lR(M) < ∞.
Also for each U 6 M , lR(M/U) 6 lR(M), lR(M/U) < ∞. We claim that
the number of non-maximal submodules of M is finite. To see this, assume
that
Tn = {X � M |lR(M/X) = n} and n0 = max{n| Card(Tn) = ∞}.
Since T1 = {X � M |lR(M/X) = 1}, then M/X is a simple R-module,
thus X is a maximal submodule of M . Hence, T1 = {X � M |X 6max
M}. By Part(1), T1 is infinite, then there exists n0 and n0 > 1. Since
lR(M/X) < lR(M) and by Remark 3.2(2), lR(M) 6 ω(Ω(M))+1. Clearly,
1 6 n0 6 ω(Ω(M)). However, since lR(M) < ∞, Theorem 5 of [8, p. 19],
implies that every proper submodule of length n0 is contained in a sub-
module of length n0 + 1. Moreover, by the definition of n0, the number of
submodules of length n0 + 1 is finite. Hence there exists a submodule N
of M such that lR(M/N) = n0 + 1 and N contains an infinite number
of submodules {Ni}i∈I of M , where lR(M/Ni)) = n0, for all i ∈ I. Now,
142 Co-intersection graph of submodules of a module
ω(Ω(M)) < ∞ implies that, there exist submodules K and L of M , with
K, L ⊆ N and lR(M/K) = lR(M/L) = n0, such that K + L = M . Since
K ∩ L ⊆ N and M/(K ∩ L) ∼= M/K ⊕ M/L,
n0 + 1 = lR(M/N) > lR(M/(K ∩ L)) = lR(M/K ⊕ M/L)
= lR(M/K) + lR(M/L) = 2n0.
Then n0 = 1. Thus, only T1 is infinite. Consequently, the number of
non-maximal submodules of M is finite.
(3) In order to establish this Part, if ω(Ω(M)) = 1, there is nothing
to prove. Let ω(Ω(M)) > 1. Since, the sum of two distinct maximal
submodules is equal M , they are not two adjacent vertices of Ω(M). Now,
by Part (1), the number of maximal submodules of M is infinite. Hence,
one can color all maximal submodules by a color, and other vertices, which
are finite number, by a new color, to obtain a proper vertex coloring of
Ω(M). Therefore, χ(Ω(M)) < ∞.
(4) Suppose that S = {N 6 M |N 6max M}. Since each two elements
of S are not two adjacent vertices of Ω(M), then S is an independent set of
the graph Ω(M). By Part (1), Card(S) = ∞. Hence, α(Ω(M)) = ∞.
4. Conclusions and future work
In this work we investigated many fundamental properties of the graph
Ω(M) such as connectivity, the diameter, the girth, the clique number,
the chromatic number and obtain some interesting results with finiteness
conditions on them. However, in future work, shall search the supplement
of this graph and research on deeper properties of them.
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Contact information
L. A. Mahdavi,
Y. Talebi
Department of Mathematics, Faculty of Mathe-
matical Sciences,
University of Mazandaran, Babolsar, Iran
E-Mail(s): l.mahdavi@stu.umz.ac.ir
talebi@umz.ac.ir
Received by the editors: 21.10.2013
and in final form 12.09.2015.
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