On the intersection graph of ideals of a commutative ring
F. Heydari




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On the intersection graph of ideals of a commutative ring
Keywords: Intersection graph, perfect graph, clique number, chromatic number, diameter, girth.   2010 Mathematics Subject Classification: 05C15, 05C17, 05C69, 13A99, 13C99.

    F. Heydari

Department of
 Mathematics, Karaj Branch,
Islamic Azad University, Karaj, Iran

f-heydari@kiau.ac.ir 

    
=======================================================================================================================================================================================================================================




Let R be a commutative ring and M be an R-module, and let I(R)^* be the set of all non-trivial ideals of R.
The M-intersection graph of ideals of R, denoted by G_M(R),
is a graph with the vertex set I(R)^*, and two distinct vertices I and J are adjacent if and only if
IM∩ JM≠ 0. For every multiplication R-module M, the diameter and the girth of G_M(R) are determined. Among other results, we prove that if M is a faithful R-module and
the clique number of G_M(R) is finite, then R is a semilocal ring. We denote the ℤ_n-intersection graph of ideals of the ring ℤ_m by G_n(ℤ_m), where n,m≥ 2 are integers and ℤ_n is a ℤ_m-module. We determine the values of n and m for which G_n(ℤ_m) is perfect. Furthermore, we derive a sufficient condition for G_n(ℤ_m) to be weakly perfect.





§ INTRODUCTION

Let R be a commutative ring, and I(R)^* be the set of all non-trivial ideals of R.
There are many papers on assigning a graph to a ring R, for instance see [1–4].
Also the intersection graphs of some algebraic structures such as groups, rings and modules have been studied by several authors, see <cit.>.
 In <cit.>, the intersection graph of ideals of R, denoted by G(R), was introduced as the graph with vertices I(R)^* and for distinct I,J∈ I(R)^*,
 the vertices I and J are adjacent if and only if I∩ J≠ 0. Also in <cit.>, the intersection
graph of submodules of an R-module M, denoted by G(M), is defined to be the graph whose vertices are
the non-trivial submodules of M and two distinct vertices are
adjacent if and only if they have non-zero intersection.
 In this paper, we generalize G(R) to G_M(R), the M-intersection graph of ideals of R, where M is an R-module.

Throughout the paper, all rings are commutative with non-zero identity and all modules are unitary.
A module is called a uniform module if the intersection of any two non-zero submodules is non-zero. An R-module M is said to be a multiplication module if every submodule of M is of the form IM, for some ideal I of R.
The annihilator of M is denoted by ann(M). The module M is called a faithful R-module if ann(M)=0. By a non-trivial submodule of
M, we mean a non-zero proper submodule of M. Also, J(R) denotes
the Jacobson radical of R and Nil(R) denotes the ideal of all nilpotent elements of R. By Max(R), we denote the set of all maximal ideals of R.

A ring having only finitely many maximal ideals is said to be a semilocal ring.
As usual, ℤ and ℤ_n will denote the integers and the integers modulo n, respectively.

A graph in which any two distinct vertices are adjacent is called a complete graph. We denote the complete graph on n vertices by K_n. A null graph is a graph containing no edges. Let G be a graph. The complement of G is denoted by G. The set of vertices and the set of edges of G are
denoted by V(G) and E(G), respectively. A subgraph H of G is said to be an induced subgraph of G if it has exactly the edges that appear
in G over V(H). Also, a subgraph H
of G is called a spanning subgraph if V(H)=V(G). Suppose that x,y∈ V(G). We denote by deg(x) the degree of a vertex x in G. A regular graph is a graph where each vertex has the same degree. We recall
that a walk between x and y is a sequence x=v_0 — v_1 — ⋯ — v_k=y of vertices of G such that for every i with 1≤ i ≤ k,
the vertices v_i-1 and v_i are adjacent. A path between x and y is a walk between x and y without repeated vertices. We say that G is
connected if there is a path between any two distinct
 vertices of G.
For vertices x and y of G, let d(x,y) be the length of a shortest path from x to y (d(x,x)=0 and d(x,y)=∞ if there is no path between x
and y). The diameter of G, diam(G), is the supremum of the set {d(x,y) : x  and y  are vertices of G}.
The girth of G, denoted by gr(G), is the length of a shortest cycle in G (gr(G)=∞ if G contains no cycles).
A clique in G is a set of pairwise adjacent vertices and the number of vertices in the largest clique of G,
denoted by ω(G), is called the clique number of G. The chromatic number of G, χ(G), is the minimal number of colors which can be assigned to the vertices of G in such a way that every two adjacent vertices have different colors. A graph G is perfect if for every induced subgraph H of G, χ(H)=ω(H). Also, G is called weakly perfect if χ(G)=ω(G).

In the next section, we introduce the M-intersection graph of ideals of R, denoted by G_M(R), where R is a commutative ring and M is a
non-zero R-module. It is shown that for every multiplication R-module M, diam(G_M(R))∈{0,1,2,∞} and gr(G_M(R))∈{3,∞}. Among other results, we prove that if M is a faithful R-module and
ω(G_M(R)) is finite, then |Max(R)|≤ω(G_M(R))+1 and J(R)=Nil(R). In the last section, we consider the ℤ_n-intersection graph of ideals of ℤ_m, denoted by G_n(ℤ_m), where n,m≥ 2 are integers and ℤ_n is a ℤ_m-module. We show that G_n(ℤ_m) is a perfect graph if and only if n has at most four distinct prime divisors. Furthermore, we derive a sufficient condition for G_n(ℤ_m) to be weakly perfect. As a corollary, it is shown that the intersection graph of ideals of ℤ_m is weakly perfect, for every integer m≥ 2.



§ THE M-INTERSECTION GRAPH OF IDEALS OF R


In this section, we introduce the M-intersection graph of ideals of R and study its basic properties.


Definition.
Let R be a commutative ring and M be a non-zero R-module. The M-intersection graph of ideals of R, denoted by G_M(R), is the
graph with vertices I(R)^* and two distinct vertices I and J are adjacent if and only if IM∩ JM≠ 0.



Clearly, if R is regarded as a module over itself, that is, M=R, then the M-intersection graph of ideals of R is exactly the same as the intersection
graph of ideals of R. Also, if M and N are two isomorphic R-modules, then G_M(R) is the same as G_N(R).


Let R=ℤ_12. Then we have the following graphs.



    
        [vstyle=Classic]
        [x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=4ℤ_12]4
        [x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=2ℤ_12]2
        [x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=6ℤ_12]6
        [x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=3ℤ_12]3
        (2,3)
        (6,3)
        (2,6)
        (2,4)

        
    

    
        [vstyle=Classic]
        [x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=4ℤ_12]4
        [x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=2ℤ_12]2
        [x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=6ℤ_12]6
        [x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=3ℤ_12]3

        
    

    
        [vstyle=Classic]
        [x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=4ℤ_12]4
        [x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=2ℤ_12]2
        [x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=6ℤ_12]6
        [x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=3ℤ_12]3
        (2,4)


        
    

    
        [vstyle=Classic]
        [x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=4ℤ_12]4
        [x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=2ℤ_12]2
        [x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=6ℤ_12]6
        [x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=3ℤ_12]3
        (2,3)
        (6,3)
        (2,6)


        
    
 


G(ℤ_12) 
G_ℤ_2(ℤ_12) 
G_ℤ_3(ℤ_12) 
G_ℤ_4(ℤ_12)





Let n≥ 2 be an integer. If [m_1,m_2] is the least common multiple of two distinct integers m_1,m_2≥ 2, then
m_1ℤℤ_n∩ m_2ℤℤ_n=m_1ℤ_n∩ m_2ℤ_n=[m_1,m_2]ℤ_n.
Thus m_1ℤ and m_2ℤ are adjacent in G_ℤ_n(ℤ) if and only if n does not divide [m_1,m_2].
 



Let p be a prime number and n,m be two positive integers. If p^n divides m, then mℤ is an isolated vertex of G_ℤ_p^n(ℤ).
Therefore, since ℤ_p^n is a uniform ℤ-module, so
G_ℤ_p^n(ℤ) is a disjoint union of an infinite complete graph and its complement.
Also, ℤ_p^∞ (the quasi-cyclic p-group),
is a uniform ℤ-module and ann(ℤ_p^∞)=0. Hence G_ℤ_p^∞(ℤ) is an infinite complete graph.






Obviously, if M is a faithful multiplication R-module, then G_M(R) is a complete graph if and only if M is
a uniform R-module.





Let R be a commutative ring and let M be a non-zero R-module.



  * If M is a faithful R-module, then G(R) is a spanning subgraph of G_M(R). To see this, suppose that I and J are adjacent vertices
of G(R). Then I∩ J≠ 0 implies that (I∩ J)M≠ 0 and so IM∩ JM≠ 0. Therefore I is adjacent to J in G_M(R).


  * If M is a multiplication R-module, then G(M) is an induced subgraph of G_M(R). Note that for each non-trivial submodule N of M,
there is a non-trivial ideal I of R, such that N=IM and so we can assign N to I. Also, N=IM is adjacent to K=JM in G(M) if and only if
IM∩ JM≠ 0, that is, if and only if I is adjacent to J in G_M(R).








Let R be a commutative ring and let M be a faithful R-module. If G_M(R) is not connected, then M is a direct sum of two R-modules.



Suppose that C_1 and C_2 are two distinct components of
G_M(R). Let I∈ C_1 and J∈ C_2. Since M is a faithful R-module, so IM∩ JM=0 implies that
I⊈ J and J⊈ I. Now if I+J≠ R, then I — I+J — J is a path between I and J, a contradiction. Thus I+J= R and so
M=IM⊕ JM.


The next theorem shows that for every multiplication R-module M, the diameter of G_M(R) has 4 possibilities.



Let R be a commutative ring and M be a multiplication R-module. Then diam(G_M(R))∈{0,1,2,∞}.



Assume that G_M(R) is a connected graph with at least two vertices. So M is a faithful module. If there is a non-trivial ideal I of R such that IM=M, then I is adjacent to all other vertices. Hence diam(G_M(R))≤ 2. Otherwise, we claim that G(M) is connected.
Let N and K be two distinct vertices of G(M). Since M is a multiplication module, so N=IM and K=JM, for some non-trivial ideals
I and J of R. Suppose that I=I_1 — I_2 — ⋯ — I_n=J is a path between I and J in G_M(R). Therefore,
N — I_2M — ⋯ — I_n-1M — K is a walk between N and K. Thus, we conclude that there is also a path between N and K in G(M).
The claim is proved. So by <cit.>, diam(G(M))≤ 2. Now, suppose that I_1 and I_2 are two distinct vertices of G_M(R). If I_1M∩ I_2M=0, then I_1M and I_2M are two distinct vertices of G(M). Hence there exists a non-trivial submodule N of M which is adjacent to both I_1M and I_2M in G(M). Since M is a multiplication module, so N=JM, for some non-trivial ideal J of R. Thus J is adjacent to both I_1 and I_2 in G_M(R). Therefore diam(G_M(R))≤ 2.




Let R be a commutative ring and M be a multiplication R-module. If G_M(R) is a connected regular graph of finite degree, then G_M(R) is a complete graph.



Suppose that G_M(R) is a connected regular graph of finite degree. If ann(M)≠ 0, then G_M(R)=K_1. So assume that ann(M)=0. We claim that M is an Artinian module. Suppose to the contrary that M is not an Artinian module. Then there is a
descending chain I_1M⊃ I_2M⊃⋯⊃ I_nM⊃⋯ of submodules of M, where I_i's are non-trivial ideals of R. This implies that deg(I_1) is infinite, a contradiction. The claim is proved. Therefore M has at least one minimal submodule. To complete the proof, it suffices to show that M contains a unique minimal submodule. By contrary, suppose that N_1 and N_2 are two distinct minimal submodules of M. Hence N_1=I_1M and N_2=I_2M, where I_1 and I_2 are two non-trivial ideals of R. Since N_1∩ N_2=0, so I_1 and I_2 are not adjacent. By Theorem <ref>, there is a vertex J which is adjacent to both I_1 and I_2. So both
I_1M and I_2M are
contained in JM. Thus each vertex adjacent to I_1 is adjacent to J too. This implies that
deg(J) > deg(I_1), a contradiction.




Also, the following theorem shows that for every multiplication R-module M, the girth of G_M(R) has 2 possibilities.


 Let R be a commutative ring and M be a multiplication R-module. Then gr(G_M(R))∈{3,∞}.




Suppose that I_1 — I_2 — ⋯ — I_n — I_1 is a cycle of length n in G_M(R). If n=3, we are done.
Thus assume that n≥4. Since I_1M∩ I_2M≠ 0 and M is a multiplication module, we have I_1M∩ I_2M=JM, where J is
a non-zero ideal of R. If J is a proper ideal of R and J≠ I_1,I_2, then I_1 — J — I_2 — I_1 is a triangle in G_M(R).
Otherwise, we conclude that I_1M⊆ I_2M or I_2M⊆ I_1M. Similarly, we can assume that I_iM⊆ I_i+1M or I_i+1M⊆ I_iM, for every i,
1<i<n. Without loss of generality suppose that I_1M⊆ I_2M. Now, if I_2M⊆ I_3M, then I_1 — I_2 — I_3 — I_1 is a cycle
of length 3 in G_M(R). Therefore assume that I_3M⊆ I_2M. Since I_3M⊆ I_4M or I_4M⊆ I_3M, so I_2 — I_3 — I_4 — I_2 is a triangle in G_M(R). Hence if G_M(R) contains a cycle, then gr(G_M(R))=3.







 Let R be a commutative ring and M be a non-zero R-module. If I is an isolated vertex of G_M(R), then the following hold:


  * I is a maximal ideal of R or I⊆ ann(M).


  * If I⊈ ann(M), then I=Ra, for every a∈ I∖ ann(M).






 (1) There is a maximal ideal 𝔪 of R such that I⊆𝔪. Assume that I≠𝔪. Then we have
IM=IM∩𝔪M=0, since I is an isolated vertex. So I⊆ ann(M).

(2) Suppose that a∈ I∖ ann(M) and I≠ Ra. Since I is an isolated vertex, we have RaM=IM∩ RaM=0 and so a∈ ann(M), a contradiction.
Thus I=Ra.





Let R be a commutative ring and M be a faithful R-module. If G_M(R) is a null graph, then it has at most two vertices and R is isomorphic to one of the
following rings:


  * F_1× F_2, where F_1 and F_2 are fields;


  * F[x]/(x^2), where F is a field;


  * L, where L is a coefficient ring of characteristic p^2, for some prime number p.




By Lemma <ref>, every non-trivial ideal of R is maximal and so by <cit.>, R cannot have more than two different
non-trivial ideals. Thus G_M(R) has at most two vertices. Also, by <cit.>, R is isomorphic to one of the
mentioned rings.



In the next theorem we show that if M is a faithful R-module and
ω(G_M(R))<∞, then R is a semilocal ring.


 Let R be a commutative ring and M be a faithful R-module. If ω(G_M(R)) is finite then
|Max(R)|≤ω(G_M(R))+1 and J(R)=Nil(R).



First we prove that |Max(R)|≤ω(G_M(R))+1. Let ω=ω(G_M(R)). By contradiction, assume that
𝔪_1,…,𝔪_ω+2 are distinct maximal ideals of R. We know that 𝔪_1⋯𝔪_i≠0, for every i,
1≤ i≤ω+1. Otherwise, 𝔪_1⋯𝔪_j=0, for some j, 1≤ j≤ω+1. So
𝔪_1⋯𝔪_j⊆𝔪_j+1 and hence by Prime Avoidance Theorem <cit.>,
we have 𝔪_t⊆𝔪_j+1, for some t, 1≤ t≤ j, which is impossible. This implies that
{𝔪_1,𝔪_1𝔪_2,…,𝔪_1⋯𝔪_ω+1} is a clique in G_M(R), a contradiction.
Thus |Max(R)|≤ω+1.

Now, we prove that J(R)=Nil(R). By contrary, suppose that a∈ J(R)∖ Nil(R). Since Ra^iM∩ Ra^jM≠ 0, for every i,j, i<j and ω(G_M(R)) is finite,
we conclude that Ra^t= Ra^s, for some integers t<s. Hence a^t(1-ra^s-t)=0, for some r∈ R. Since a∈ J(R), so 1-ra^s-t is a unit.
This yields that a^t=0, a contradiction. The proof is complete.











§ THE ℤ_N-INTERSECTION GRAPH OF IDEALS OF ℤ_M


Let n,m≥ 2 be two integers and ℤ_n be a ℤ_m-module. In this section we study the ℤ_n-intersection graph of ideals of the ring ℤ_m. Also, we generalize some results given in <cit.>. For abbreviation, we denote G_ℤ_n(ℤ_m) by G_n(ℤ_m).
Clearly, ℤ_n is a ℤ_m-module if and only if n divides m. 

Throughout this section, without loss of generality, we assume that m=p_1^α_1⋯ p_s^α_s and n=p_1^β_1⋯ p_s^β_s, where p_i's are distinct primes, α_i's are positive integers, β_i's are non-negative integers, and 0≤β_i≤α_i for i=1,…,s. Let S={1,…,s} and S'={i∈ S : β _i≠ 0}. The cardinality of S' is denoted by s'.
For two integers a and b, we write a|b (a∤ b) if a divides b (a does not divide b).

First we have the following remarks.



It is easy to see that I(ℤ_m)={dℤ_m : d divides m } and
|I(ℤ_m)^*|=∏_i=1^s(α_i+1)-2. Let ℤ_n be a ℤ_m-module. If n|d, then dℤ_m is an isolated vertex of G_n(ℤ_m).
Obviously, d_1ℤ_m and d_2ℤ_m are adjacent if and only if n∤ [d_1,d_2]. This implies that G_n(ℤ_m) is a subgraph of G(ℤ_m).







Let ℤ_n be a ℤ_m-module and d=p_1^r_1⋯ p_s^r_s(≠ 1,m) be a divisor of m. We set D_d={i∈ S :  r_i< β_i }. Clearly, D_d⊆ S'. Suppose that W is a clique of G_n(ℤ_m). Then Γ_W={D_d : dℤ_m∈ W } is an intersecting family of subsets of S'. (A family of sets is intersecting if any two of its sets have a non-empty intersection.) Also, if Γ is an intersecting family of subsets of S' and W_Γ={ dℤ_m : d≠ 1,m,  d|m,  D_d∈Γ} is non-empty, then W_Γ is a clique of
G_n(ℤ_m). (If D is a non-empty subset of S' and Γ={D}, then we will denote W_Γ by W_D.) Thus we have


ω(G_n(ℤ_m))=max { |W_Γ|  :  Γ is an intersecting family of subsets of S' }.





Now, we provide a lower bound for the clique number of G_n(ℤ_m).



 Let ℤ_n be a ℤ_m-module. Then


ω(G_n(ℤ_m))≥ max {β_j∏_i≠ j(α_i+1)-1    :   β_j≠ 0 }.







 Suppose that β_j≠ 0. With the notations of the previous remark, let Γ={ D⊆ S' : j∈ D }. Then
Γ is an intersecting family of subsets of S' and so W_Γ
is a clique of G_n(ℤ_m). Clearly, |W_Γ|=β_j∏_i≠ j(α_i+1)-1.
Therefore ω(G_n(ℤ_m))≥β_j∏_i≠ j(α_i+1)-1 and hence the result holds.



Clearly, if n=p_1^β_1 (β_1>1), then equality holds in the previous theorem. Also, if n has only two distinct prime divisors, that is, s'=2, then again equality holds. So the lower bound is sharp.


Let m=n=p_1^2p_2^2p_3^2, where p_1, p_2, p_3 are distinct primes. Thus S'=S={1,2,3} and G_n(ℤ_m)=G(ℤ_m). It is easy to see that |W_{1}|=|W_{2}|=|W_{3}|=2 and |W_{1,2}|=|W_{1,3}|=|W_{2,3}|=4. Also, |W_{1,2,3}|=7. Let Γ_j={ D⊆ S' : j∈ D }, for j=1,2,3. Hence |W_Γ_j|=17, for j=1,2,3. If Γ={{1,2}, {1,3}, {2,3}, {1,2,3}}, then |W_Γ|=19. Therefore ω(G(ℤ_m))=19.



By the strong perfect graph theorem, we determine the values of n and m for which G_n(ℤ_m) is a perfect graph.



(The Strong Perfect Graph Theorem <cit.>) A finite graph G is perfect if and only if neither G nor G contains an induced odd cycle of length at least 5.



Let ℤ_n be a ℤ_m-module. Then G_n(ℤ_m) is perfect if and only if n has at most four distinct prime divisors.



First suppose that s'≥5 and n=p_1^β_1⋯ p_s'^β_s', where p_i's are distinct primes and β_i's are positive integers.
Let D_1={p_1,p_5}, D_2={p_1,p_2}, D_3={p_2,p_3}, D_4={p_3,p_4}, and D_5={p_4,p_5}. Now, assume that d_iℤ_m∈ W_D_i, for i=1,…,5. Hence d_1ℤ_m — d_2ℤ_m — d_3ℤ_m — d_4ℤ_m — d_5ℤ_m — d_1ℤ_m is an induced cycle of length 5 in G_n(ℤ_m). So by Theorem <ref>, G_n(ℤ_m) is not a perfect graph.

Conversely, suppose that G_n(ℤ_m) is not a perfect graph. Then by Theorem <ref>, we have the following cases:

Case 1. d_1ℤ_m — d_2ℤ_m — d_3ℤ_m — d_4ℤ_m — d_5ℤ_m — d_1ℤ_m is an induced cycle of length 5 in G_n(ℤ_m). Let D_i=D_d_i, for i=1,…,5. So D_5∩ D_1≠∅ and D_i∩ D_i+1≠∅, for i=1,…,4. Let p_5∈ D_5∩ D_1 and p_i∈ D_i∩ D_i+1, for i=1,…,4. Clearly, p_1,…,p_5 are distinct and thus s'≥ 5.

Case 2. d_1ℤ_m — d_2ℤ_m — d_3ℤ_m — d_4ℤ_m — d_5ℤ_m — d_6ℤ_m is an induced path of length 5 in G_n(ℤ_m). Let D_i=D_d_i, for i=1,…,6. So D_i∩ D_i+1≠∅, for i=1,…,5. Let p_i∈ D_i∩ D_i+1, for i=1,…,5. Clearly, p_1,…,p_5 are distinct and hence s'≥ 5.

Case 3. There is an induced cycle of length 5 in G_n(ℤ_m). So G_n(ℤ_m) contains an induced cycle of length 5 and by Case 1, we are done.

Case 4. d_1ℤ_m — d_2ℤ_m — d_3ℤ_m — d_4ℤ_m — d_5ℤ_m — d_6ℤ_m is an induced path of length 5 in G_n(ℤ_m). Since D_d_1∩ D_d_3≠∅, D_d_1∩ D_d_4≠∅ and D_d_3∩ D_d_4=∅, we may assume that {p_1,p_2}⊆ D_d_1, where p_1∈ D_d_3 and p_2∈ D_d_4, for some distinct p_1,p_2∈ S'. Similarly, we find that {p_3,p_4}⊆ D_d_2, for some distinct p_3,p_4∈ S'∖{p_1,p_2} and also |D_d_3|≥ 2. Now, since D_d_3∩ D_d_2=∅ and p_2∉ D_d_3, we deduce that s'≥ 5.




The graph G(ℤ_m) is perfect if and only if m has at most four distinct prime divisors.



In the next theorem, we derive a sufficient condition for G_n(ℤ_m) to be weakly perfect.



Let ℤ_n be a ℤ_m-module. If α_i≤ 2β_i-1 for each i∈ S', then G_n(ℤ_m) is weakly perfect.





Let D be a non-empty subset of S' and D=S'∖ D. As we mentioned in Remark <ref>, if W_D is non-empty, then W_D is a clique of G_n(ℤ_m). Also, the vertices of W_S' (if W_S'≠∅) are adjacent to all non-isolated vertices. Suppose that D_1 and D_2 are two non-empty subsets of S' and D_1⊆ D_2. Since α_i≤ 2β_i-1 for each i∈ S', so ∏_i∈ D_2∖ D_1(α_i-β_i+1)≤∏_i∈ D_2∖ D_1β_i. This implies that ∏_i∈ D_1β_i∏_i∉ D_1(α_i-β_i+1)≤∏_i∈ D_2β_i∏_i∉ D_2(α_i-β_i+1) and hence |W_D_1|≤ |W_D_2|. 

Let Γ be an intersecting family of subsets of S' and ω(G_n(ℤ_m))=|W_Γ|.
Let D⊆ S'. We show that D∈Γ or D∈Γ. Assume that D∉Γ. So there is D_1∈Γ such that D∩ D_1=∅. Thus D_1⊆D and hence D∈Γ.
We claim that |W_D|≤ |W_D|, for each D∈Γ. Suppose to the contrary, D∈Γ and |W_D|>|W_D|. If A∈Γ and A⊆ D, then D⊆A. So we have |W_A|≤|W_D|<|W_D|≤|W_A|. Let Φ=Γ∪{A : A∈Γ, A⊆ D}∖{A∈Γ : A⊆ D}. Then Φ is an intersecting family of subsets of S' and |W_Γ|<|W_Φ|, a contradiction. The claim is proved.

Now, we show that G_n(ℤ_m) has a proper |W_Γ|-vertex coloring. First we color all vertices of W_Γ with different colors.
Next we color each family W_D of vertices out of W_Γ with colors of vertices of W_D. Note that if D∉Γ, then D∈Γ and |W_D|≤ |W_D|. Suppose that d_1ℤ_m and d_2ℤ_m are two adjacent vertices of G_n(ℤ_m). Thus D_d_1∩ D_d_2≠∅.
Without loss of generality, one can assume
D_d_1≠ D_d_2. So we deduce that D_d_1≠D_d_2 and D_d_1≠D_d_2.
Therefore, d_1ℤ_m and d_2ℤ_m have different colors. Thus χ(G_n(ℤ_m))≤ |W_Γ| and hence ω(G_n(ℤ_m))=χ(G_n(ℤ_m))=|W_Γ|.



As an immediate consequence of the previous theorem, we have the
next result.


The graph G(ℤ_m) is weakly perfect, for every integer m≥ 2.



In the case that α_i=2β_i-1 for each i∈ S', we determine the exact value of χ(G_n(ℤ_m)). It is exactly the lower bound obtained in the Theorem <ref>.


Let ℤ_n be a ℤ_m-module. If α_i=2β_i-1 for each i∈ S', then ω(G_n(ℤ_m))=χ(G_n(ℤ_m))=2^s'-1∏_i∈ S'β_i∏_i∈ S∖ S'(α_i+1)-1.




Let D≠∅ be a proper subset of S'. Then |W_D|=∏_i∈ Dβ_i∏_i∉ D(α_i-β_i+1)=∏_i∈ S'β_i∏_i∈ S∖ S'(α_i+1) and hence |W_D|=|W_D|. Also, the vertices of W_S' (if W_S'≠∅) are adjacent to all non-isolated vertices and |W_S'|=∏_i∈ S'β_i∏_i∈ S∖ S'(α_i+1)-1. Clearly if Γ is an intersecting family of subsets of S', then |Γ|≤ 2^s'-1. Moreover, if β_j≠ 0 and Γ_j={ D⊆ S' : j∈ D }, then |Γ_j|=2^s'-1. Thus by Theorem <ref>, ω(G_n(ℤ_m))=χ(G_n(ℤ_m))=|W_Γ_j|=2^s'-1∏_i∈ S'β_i∏_i∈ S∖ S'(α_i+1)-1.




Let m=p_1⋯ p_s, where p_i's are distinct primes. Then ω(G(ℤ_m))=χ(G(ℤ_m))=2^s-1-1.




We close this article by the following problem.

Problem.
Let ℤ_n be a ℤ_m-module. Then is it true that G_n(ℤ_m) is a weakly perfect graph?







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