ON THE MODULE INTERSECTION GRAPH OF IDEALS OF RINGS

Let R be a commutative ring and M an R-module. The M -intersection graph of ideals of R is an undirected simple graph, denoted by GM (R), whose vertices are non-zero proper ideals of R and two distinct vertices are adjacent if and only if IM ∩ JM 6= 0. In this article, we focus on how certain graph theoretic parameters of GM (R) depend on the properties of both R and M . Specifically, we derive a necessary and sufficient condition for R and M such that the M -intersection graph GM (R) is either connected or complete. Also, we classify all R-modules according to the diameter value of GM (R). Further, we characterize rings R for which GM (R) is perfect or Hamiltonian or pancyclic or planar. Moreover, we show that the graph GM (R) is weakly perfect and cograph.


Introduction
Recently, there has been considerable attention in the literature to associating graphs with rings or modules. More specifically, there are many papers on assigning graphs to modules (see, for example, [3,11,13]). The present paper deals with what is known as the intersection graph. The first step in this direction was taken by Csákány and Pollák [7] in 1969. In [5], the authors introduced and studied the intersection graph of a family of non-trivial ideals of a ring. These motivated the authors of [1] to define the intersection graph of submodules of a module. In the last decade, many research articles have been published on the intersection graphs of rings and modules; for instance, see [2,4,12,14]. In 2018, Heydari [10] combined the concepts of intersection graph of ideals of a ring and intersection graph of submodules of a module to define the M -intersection graph of ideals of a ring R, where M is an R-module.
Formally, let R be a commutative ring with identity and let M be an R-module. The M -intersection graph of ideals of R, denoted by G M (R), is a simple graph whose vertices are the non-trivial ideals of R and any two distinct vertices are adjacent if IM ∩ JM = 0. Note that the properties of G M (R) depend upon M as 94 T. ASIR, A. KUMAR, AND A. MEHDI well as the underlying ring R. This observation inspired the authors of the paper to investigate further properties of G M (R).
We first summarize the notations and concepts. For any a ∈ M , ann(a) = {x ∈ R : ax = 0} is the annihilator ideal of a in M . The module M is called a faithful R-module if ann(M ) = 0. An R-module M is a multiplication module if for each submodule N of M there is an ideal I of R such that IM = N . A module is called a uniform module if the intersection of any two non-zero submodules is non-zero. Throughout the paper, the notations I(R) * , Max(R), Min(R), Z and Z n denote, respectively, the set of all non-trivial ideals of R, the set of all maximal ideals of R, the set of all minimal ideals of R, the set of integers and the set of integers modulo n.
A graph in which each pair of distinct vertices is joined by an edge is called a complete graph. On the other extreme, a null graph is a graph containing no edges. The complete graph and null graph on n vertices are denoted by K n and K n , respectively. A graph G is connected if there is a path between any two distinct vertices of G; otherwise G is disconnected. Let u and v be two distinct vertices of G. If the shortest u−v path is of length k, then d(u, v) = k. Note that d(u, v) = ∞ if there is no path between u and v. The diameter of G, denoted by diam(G), is the supremum of the set {d(u, v) : u and v are distinct vertices of G}. The girth of G, denoted by gr(G), is the length of the shortest cycle in G and gr(G) = ∞ whenever G is a tree. An Eulerian circuit in a connected graph G is a closed trail that contains every edge of G. A connected graph that contains an Eulerian circuit is called an Eulerian graph. A cycle in G that contains every vertex of G is called a Hamiltonian cycle of G. A Hamiltonian graph is a graph that contains a Hamiltonian cycle.
By a clique in G we mean a complete subgraph of G, and the number of vertices in the largest clique of G is called the clique number of G, denoted by ω(G). The chromatic number of G, denoted by χ(G), is the minimum number of colors which can be assigned to the vertices of G in such a way that every two adjacent vertices have different colors. Note that, for any graph G, ω(G) ≤ χ(G). A graph G is weakly perfect if ω(G) = χ(G) and G is called perfect if ω(H) = χ(H) for every induced subgraph H of G. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. For general references on graph theory, we use Chartrand and Zhang [6].
The article is organized as follows. In Section 2, we obtain a necessary and sufficient condition for the connectedness and completeness of G M (R), followed by diameter classification of G M (R). In Section 3, we provide an equivalent condition for G M (R) to be a tree and also characterize all Noetherian rings R with unique minimal ideal for which G M (R) is perfect. Finally, in Section 4 we concentrate on the cyclic nature of G M (R). In particular, we obtain the girth value of G M (R) and discuss about Hamiltonian and pancyclic nature of G M (R).
We begin with the observation that there are R-modules for which G M (R) satisfies the extreme cases, namely null graph and complete graph. If M = Z p ⊕ Z q and R = Z pq for distinct primes p and q, then G M (R) is a null graph. Further, the following result says that G M (R) is complete whenever M is a uniform faithful R-module. Proposition 1.1. Let R be a commutative ring and let M be a uniform R-module.
This implies that A is the set of isolated vertices in G M (R). Also for any two ideals J, K ∈ I(R) * \ A, we have JM and KM as non-zero submodules of M . Since M is uniform, JM ∩ KM = 0. This implies that the subgraph induced by I(R) * \ A is complete in G M (R). Moreover, if M is faithful, then ann(M ) = 0 and so there is no ideal I in R such that IM = 0. Thus A = ∅.

Connectedness
Connectedness is one of the significant graph theoretic properties. In this section, we investigate conditions for which the graph G M (R) is connected. The main result of this section is Theorem 2.7, in which we classify the modules according to the diameter of G M (R). In order to prove Theorem 2.7, we need a few propositions and lemmas.
Notice that if a module M is not faithful, then ann(M ) = (0) and ann(M ) ∈ I(R) * . Also ann(M )M = (0) so that ann(M ) is an isolated vertex of G M (R). Therefore G M (R) is disconnected. In case of a faithful module, Heydari [10] provides a necessary condition for the disconnected G M (R) which is stated below. In the following theorem we observe a few interesting consequences of the above result. In what follows, for a given R-module M , G(M ) denotes the intersection graph of M (see [1]). (ii) Assume that |I(R) * | ≥ 2, G M (R) is disconnected and G M (R) is not a null graph. We prove that every ideal of R is minimal. Let I be an arbitrary ideal in R. Let C 1 and C 2 be two components of G M (R). Without loss of generality, assume that I ∈ C 1 . Choose an ideal J ∈ C 2 . If I + J = R, then I → I + J → J is a path in G M (R), a contradiction. Thus I + J = R. Since I and J are not adjacent in G M (R), IM ∩ JM = 0. Consequently, I ∩ J = 0. We claim that I is a minimal ideal of R. Suppose K is an ideal such that K I. Clearly K ∈ C 1 . By the above argument, we get K +J = R and K ∩J = 0. Let x ∈ I. Since J +K = R, x = y +z for some y ∈ J and z ∈ K. Since y = x − z ∈ I and I ∩ J = 0, we have x − z = 0 and so x = z ∈ K. Therefore I = K, a contradiction. Thus I is a minimal ideal in R. Equivalently any ascending (or descending) chain of ideals has exactly two non-zero ideals.
The next result characterizes all modules for which the graph G M (R) is not connected. In this regard, notice that Chakrabarty et al. [5] characterized all disconnected intersection graphs of ideals of a ring.

Theorem 2.3. Let R be a commutative ring and M a multiplication R-module. Then G M (R) is disconnected if and only if either M is not faithful or every nontrivial ideal of R is minimal and
Assume M is faithful. Let I and J belong to different components, say I ∈ C 1 and J ∈ C 2 , where C 1 and C 2 are two distinct components of G M (R). Then IM ∩ JM = 0 so that I ∩ J = 0. Clearly which is a contradiction as I and J are in different components. Let us take i ∈ I.
Now we write the other version of the above result as follows. This implies that I∈I(R) * I = 0 and it is the minimal ideal of R, which is unique. It remains to prove that R is Artinian. Suppose that, on the contrary, R is not Artinian. Let I be the unique minimal ideal of R. Then there exists a chain of ideals J 1 ⊇ J 2 ⊇ · · · J n ⊇ · · · in R which does not contain the minimal ideal I. So I J n for all n ∈ Z + . Therefore J n ∩ I = 0 for every n. That is, I is not adjacent to any ideal J n in the chain and G M (R) is not complete.
The following points are worth to mention in the context of Theorem 2.5.
any two maximal ideals have non-zero intersection. Therefore, either I or J is not maximal; say I is not maximal. Now, there exists an ideal K such that I ⊂ K ⊂ R.
The next result gives the structure of the particular graph, namely G Zn (Z). Let I k , I ∈ A with I k Z n ∩ I Z n = 0. Then choose I t ∈ A such that (t, n) = 1. This implies that I t Z n = Z n so that I k → I t → I is a path in G Zn (Z). Therefore the subgraph induced by A is connected.

Perfectness
The theory of perfect graphs relates the concept of graph colorings to the concept of cliques. The study of perfect graphs is very significant because a number of important algorithms only work on perfect graphs and perfect graphs can be used in a wide variety of applications, ranging from scheduling to order theory to communication theory. Note that it is sufficient to prove that a graph G is perfect if and only if it does not contain an odd cycle of length greater than or equal to 5.
In this section, we investigate whether G M (R) is perfect or not. We start with the following remark on the clique number of G M (R).  Then the following conditions are equivalent: Similarly, I 1 ∩ I 2 is adjacent to I 2 . Therefore I 1 → I 1 ∩ I 2 → I 2 → I 1 so that G M (R) contains a cycle of length 3, contradicting the fact that χ(G M (R)) ≤ 2. Therefore R is local.
Let I be the unique maximal ideal of R and let J be an arbitrary ideal of R. Since every ideal is contained in I, we have J ⊂ I. If J contains an ideal K, then K → J → I → K is a cycle of length 3, a contradiction. Thus J is a minimal ideal of R and equivalently the length of the module is at most 3. Moreover, since M is a faithful multiplication module with finite length, we have R ∼ = M and so G(M ) ∼ = G M (R). Next, we obtain a necessary and sufficient condition for the perfectness of G M (R) when R is a direct product of Noetherian rings, each of which has a unique minimal ideal. Proof. (⇒): Suppose n ≥ 5. Let I j = (0) × · · · × (0) × R j × R j+1 × (0) × · · · × (0) for j = 1, 2, 3, 4 and We claim that every odd cycle of length more than 4 in G M (R) must have diagonals. In order to prove the claim, suppose I 1 → I 2 → I 3 → · · · → I m → I 1 is a cycle of odd length m ≥ So let n = 1. If the length of M is less than 4, then by Theorem 3.2, G M (R) is a tree and so it is perfect. Therefore the length of M is at least 4. Since M is faithful, the subgraph induced by any chain of ideals of R is complete in G M (R). Thus G M (R) is perfect.

Theorem 3.3. Let
As we have seen, the M -intersection graphs of modules are not perfect in general. So we investigate whether they are weakly perfect, and the answer is yes. The next result determines the value of ω(G M (R)) and χ(G M (R)) when M is the direct sum of finite simple modules.
Define V 1 = {I ∈ I(R) * : IM ∈ A 1 } and V i = {I ∈ I(R) * : IM ∈ A i } for i = 2, . . . , n. Clearly I(R) * = n+1 k=0 V k and V k 's are mutually disjoint. Note that every vertex of V 0 is isolated in G M (R) and the subgraph induced by V k for k = 1, . . . , n + 1 is complete in G M (R). For 1 ≤ i ≤ n, let I ∈ V i and J ∈ V n+1 . Then IM ∩ JM ⊇ M i = {0}. Therefore every vertex in V n+1 is adjacent to all the vertices of V i . Further, let K ∈ V j for some 1 ≤ i = j ≤ n. Since M is a multiplication module, there exists an ideal L in R such that LM = M i . Since M i is simple, L is not adjacent to K in G M (R). Therefore, in G M (R), every vertex of Let us close this section by applying the above result for finding the clique and coloring number of specific M -intersection graphs.

Cyclic subgraph and planarity
In this section, we discuss about some cyclic substructure and planarity of G M (R). Let us start with the girth value of G M (R). In [10, Theorem 4], Heydari determined the girth of G M (R) in case of a multiplication R-module M . We now generalize it. The next result helps us to find the length of the largest induced cycle and induced path in G M (R). The length of the longest induced path in G is called the induced detour number of G, denoted by idn(G), and the maximum length of an induced cycle in G is called the induced circumference of G, denoted by icir(G); see [9].
. . , n − 1 and I n = R 1 ×(0)×· · ·×(0)×R n . For a fixed k, 3 ≤ k ≤ n, take J k = R 1 ×(0)×· · ·×(0)×R k × (0) × · · · × (0). Now the subgraph induced by the vertices {I 1 , . . . , I k−1 , J k } forms a cycle of length k in C M (R). So C n is an induced subgraph of G M (R). Further, a similar proof technique as in Theorem 3.3 leads us to say that every cycle of length more than n in G M (R) must have a diagonal. Thus icir(G M (R)) = n. Also, in this case, it is equivalent to saying that idn(G M (R)) = n − 1.
Other important concepts related with the cyclic structure of a graph are those of being Eulerian or Hamiltonian. First of all note that if R = I + J for any I, J ∈ I(R) * , then we can have a cycle I 1 → I 1 + I 2 → I 2 + I 3 → · · · → I n−1 + I n → I n + I 1 → I 1 of length n + 1 in G M (R), where I 1 , . . . , I n ∈ I(R) * . In this regard, we add a couple of observations.
. Note that u j = v j for all j = 1, . . . , n − 2 and the subgraph induced by the set {u 1 , . . . , u n−2 , v 1 , . . . , v n−2 } in G M (R) is complete. Now, in G M (R), start a path P from the vertex u 1 ∈ A 1 and travel along all the vertices of A 1 and end up in v 1 ∈ A 1 . Since v 1 is adjacent to u 2 ∈ A 2 , continue the path P to u 2 , then travel all the vertices of A 2 and end up in v 2 ∈ A 2 . Continuing this process n − 2 steps, we get the path P containing all the vertices of n−2 j=1 A j which end up in v n−2 . Notice that the vertex v n−2 is adjacent to all the vertices of A n−1 in G M (R). So now extend the path P to any vertex of A n−1 and then travel along all the vertices of A n−1 and arrive at the vertex (0) × · · · × (0) × K n−1 × R n ∈ A n−1 . Since the vertex (0) × · · · × (0) × K n−1 × R n is adjacent to all the vertices of A n , by repeating the process to the vertices of A n , we get P as a Hamiltonian path which ends at the vertex (0) × · · · × (0) × R n ∈ A n . Now the edge between (0) × · · · × (0) × K n−1 × R n and (0) × · · · × (0) × R n leads to a Hamiltonian cycle in G M (R).
Suppose n = 2. There are three possibilities: (1) both R 1 and R 2 are not fields, (2) one of R 1 or R 2 is a field and the other is not a field, or (3) both R 1 and R 2 are fields.
Case 1: Assume both R 1 and R 2 are not fields. Let K 1 ∈ I(R 1 ) * and K 2 ∈ I(R 2 ) * . Here start a path Q from K 1 × R 2 ∈ A 1 , travel along all the vertices of A 1 and end at K 1 × K 2 ∈ A 1 . Then move the path Q to the vertex (0) × R 2 ∈ A 2 and move on to all the vertices of A 2 and finally end at (0) × K 2 ∈ A 1 . Now the path Q together with the edge between (0) × K 2 and K 1 × R 2 forms a Hamiltonian cycle in G M (R).
Case 2: Assume that R 1 is a field and R 2 is not a field. If K 21 , K 22 ∈ I(R 2 ) * with K 21 = K 22 , then start a path Q from R 1 × K 21 ∈ A 1 , travel along all the vertices of A 1 and end at R 1 ×K 22 ∈ A 1 . Then move into A 2 through the vertex (0)×K 21 ∈ A 2 and travel all the vertices of A 2 and end at up (0) × R 2 ∈ A 1 . Now the path Q together with the edge between (0)×R 2 and R 1 ×K 21 serves as a Hamiltonian cycle in G M (R). If I(R 2 ) * = {K 2 }, then I(R) * = {R 1 × (0), R 1 × K 2 , (0) × K 2 , (0) × R 2 }. Therefore the vertex R 1 × (0) is an end vertex in G M (R) so that G M (R) does not contain a cycle. Thus, in this case, |I(R 2 ) * | = 1.
Case 3: If R 1 and R 2 are fields, then |I(R) * | = 2, a contradiction. and the vertex (0) × R 2 × · · · × R n is adjacent to all the vertices of A in G M (R). Thus K 2 n is a subgraph of G M (R).
To end this section, recall that every Artinian ring can be decomposed into local rings. So the following result is valid for all Artinian rings. Suppose n = 2. There are three possibilities: Case 1: Assume both R 1 and R 2 are not fields. Let K 1 ∈ I(R 1 ) * and K 2 ∈ I(R 2 ) * . Then the subgraph induced by the vertex subset {K 1 × (0), K 1 × K 2 , K 1 × R 2 , R 1 × (0), R 1 × K 2 } is K 5 , a contradiction.