The intersection graph of graded submodules of a graded module


               <jats:p>In this article, we introduce and study the intersection graph of graded submodules of a graded module. Let <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>M</m:mi>
                        </m:math>
                        <jats:tex-math>M</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> be a left <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_002.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>G</m:mi>
                        </m:math>
                        <jats:tex-math>G</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-graded <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_003.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>R</m:mi>
                        </m:math>
                        <jats:tex-math>R</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-module. We define the intersection graph of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_004.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>G</m:mi>
                        </m:math>
                        <jats:tex-math>G</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-graded <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_005.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>R</m:mi>
                        </m:math>
                        <jats:tex-math>R</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-submodules of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_006.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>M</m:mi>
                        </m:math>
                        <jats:tex-math>M</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, denoted by <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_007.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi mathvariant="normal">Γ</m:mi>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>G</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>R</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>M</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>\Gamma \left(G,R,M)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, to be a simple undirected graph whose set of vertices consists of all nontrivial <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_008.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>G</m:mi>
                        </m:math>
                        <jats:tex-math>G</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-graded <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_009.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>R</m:mi>
                        </m:math>
                        <jats:tex-math>R</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-submodules of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2022-0005_eq_010.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>M</m:mi>
                        </m:math>
                        <jats:tex-math>M</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, where two vertices are adjacent if their intersection is nonzero. We study properties of these graphs, such as connectivity, diameter, and girth. We also investigate the intersection graph of graded submodules for certain types of gradings such as faithful and strong gradings.</jats:p>


Introduction
Studies of graphs associated with algebraic structures developed remarkably in recent years. Usually, the purpose of associating a graph with an algebraic structure is to investigate the algebraic properties using concepts in graph theory. Zero-divisors graph, total graphs, annihilating-ideal graph, and unit graphs are very interesting examples of graphs associated with rings, see [1][2][3][4]. For studies on graphs associated with graded rings and graded modules, in particular, see [5,6].
Among the types of graphs associated with rings are intersection graphs. In 2009, Chakrabarty et al. [7] introduced and studied the intersection graph of ideals of a ring R, which is an undirected simple graph, denoted by ( ) G R , whose vertices are the nontrivial left ideals of R and two vertices I and J are adjacent if their intersection is nonzero. Inspired by their work, Akbari et al. [8] introduced the intersection graph of submodules of a module. For a ring R with unity and a unitary left R-module M, the set of all R-submodules of M is denoted by ( ) * S M . The intersection graph of submodules of M, denoted by ( ) G M , is an undirected simple graph defined on ( ) * S M , where two non-trivial submodules are adjacent if they have a nonzero intersection. Since they were introduced, intersection graphs of ideal and submodules have attracted many researchers to study their graph-theoretic properties and investigate their structures (see [9][10][11][12][13][14][15][16][17]). Alraqad et al. [18] introduced and studied the intersection graph of graded ideals of a graded ring.
Motivated by all previous works, we introduce the intersection graph of graded submodules of a graded module. Let  , is defined to be an undirected simple graph whose set of vertices is ( ) * hS M and two vertices N and K are adjacent if We aim to study the properties of these graphs analogous to the nongraded case. In addition, we investigate connections and relationships among ( ) , and ( ) G M under certain types of gradings.
The organization of the paper is as follows: Section 2 is devoted to the study of graph-theoretic properties of ( ) . We discuss their connectivity, diameter, regularity, completeness, domination numbers, and girth. In Section 3, we investigate the relationships between ( ) G R M Γ , , and ( ) G M σ (where M σ is considered as a left R e -module) under some types of gradings such as faithful grading and strong grading. This section also presents some results regarding the relationship between ( ) G R M Γ , , and ( ) G M when the grading group is a linearly ordered group.
For standard terminology and notion in the graph theory, we refer the reader to the textbook [19]. For a simple graph, Γ, the set of vertices and set of edges are denoted by ( ) . The graph Γ is said to be regular if all of its vertices have the same degree. A graph is called complete (resp. null) if any pair of its vertices are adjacent (resp. not adjacent). A complete (resp. null) graph with n vertices is denoted by K n (resp. N n ). A graph is said to be connected if any pair of its vertices is connected by a path.
Throughout this article, all rings are associated with unity ≠ 1 0, and all modules are left modules. When a ring R is G-graded, we denote that by ( A G-graded R-submodule of M is said to be G-graded maximal (resp. simple or minimal) if it is maximal (resp. minimal) among all proper (resp. nonzero) G-graded R-submodules of M. We denote by ) the set of all nontrivial G-graded maximal (resp. simple) R-submodules.
. We say that M is G-graded left Noetherian (resp. Artinian) if M satisfies the ascending (resp. descending) chain condition for the G-graded R-submodules of M.

Graph theoretic properties of G R M Γ , , ( )
We present the following well-known technical lemma in this section.
The following two results from [8] classify disconnected intersection graphs of submodules. Analogues to the nongraded case, next we characterize disconnected intersection graphs of graded submodules.
Theorem 2.4. Let R be a G-graded ring and M be a G-graded R-module such that | ( )| ≥ G R M Γ , , 2. Then, the followings are equivalent: is a null graph.
is disconnected. For a contradiction, assume N and K are two adjacent vertices. So N , K , and ∩ N K belong to the same component of ( ) is disconnected, there is a vertex L that is not connected to any of the vertices N , K , and contains no edges, and hence, it is a null graph.
4 Let N and K be G-graded maximal as well as G-graded simple R-submodules of M. Then, is disconnected. are distinct. The next theorem presents an obvious relation between these two graphs.
is disconnected if and only if G M is a complete graph with at least two vertices.

Proof. The result follows by Theorem 2.4 and [6, Theorem 2.2]. □
The proof of the next corollary is straightforward.
is connected, then every pair of G-graded maximal R-submodules intersect nontrivially.
, between any two vertices x y , in a graph Γ is the length of the shortest path between them, and . Thus, we have the path ↔ ⊕ ↔ N L K K, and hence, ( is not a null graph. Then, the followings are equivalent: is regular. Assume that M contains two distinct G-graded simple R-submodules N and K . Clearly, N and K are nonadjacent. By Theorem 2.7, there is a G-graded R-submodule Y that is adjacent to both N and K . Hence, by minimality of N , we obtain is complete, then every G-graded R-submodule is G-essential, and thus, by Theorem 2.8, Gsoc . Recall that the girth of a graph Γ, denoted by ( ) g Γ , is the length of its shortest cycle. If Γ has no cycles, then ( ) = ∞ g Γ . 4. This implies that every pair of distinct nontrivial G-graded submodules of M with nonzero intersection should be comparable, otherwise ( ) will have a cycle of length 3, a contradiction. Since ( ( contains a path of length 3, say ↔ ↔ ↔ N L K P. Since any two submodules in this path are comparable and any chain of nontrivial G-graded submodules of length 2 induces a cycle of length 3 in ( ) , the only possible two cases are ⊆ N L, ⊆ K L, ⊆ K P or ⊆ L N , ⊆ L K , ⊆ P K . The first case yields ⊆ ∩ K L P, and hence, , a contradiction. In the second case, we have , which again yields a contradiction. Therefore, ( ( The next theorem gives a characterization of G-graded R-modules such that ( ( Recall that a graph is called star if it has no cycles and has one vertex (the center) that is adjacent to all other vertices.
is not a null graph with is a star graph whose center is the unique G-graded maximal R-submodule of M.
is connected. Suppose that N 1 and N 2 are two distinct G-graded maximal R-submodules of M. By Theorem 2.7, is null, which contradicts the assumption that ( ) is not null. Therefore, M is G-graded local module. Let N be the G-graded maximal submodule of M. It is easy to see that every proper graded submodule of a G-graded Noetherian module is contained in a G-graded maximal submodule. So has no cycles, we conclude that ( ) is a star graph. □ A subgraph ϒ of a graph Γ is called an induced subgraph if any edge in Γ that joins two vertices in ϒ is in ϒ. A complete induced subgraph of a graph Γ is called a clique, and the order of the largest clique in Γ, denoted by ( ) ω Γ , is the clique number of Γ.
, then M is G-graded Artinian and G-graded Noetherian.
Proof. Members of any ascending or descending chain of G-graded R-submodules form a clique in , and hence, the chain is finite. □ . In this result, we use the notion of graded decomposable modules. A G-graded R-module M is called G-graded decomposable, if it is a direct sum of two nontrivial G-graded R-submodules. If M is not G-graded decomposable, then it is called G-graded indecomposable.
Theorem 2.14. Let M be a G-graded R-module that contains a G-graded maximal submodule. Then,  , < k 0. The -graded 6 -submodules 6 and + x 6 6 are adjacent in (  [] ) x Γ , , 6 6 , and by Theorem 2.4, we have (   [] ) x Γ , , 6 6 is connected. On the other hand, for each ≥ k 0, ⟨ ⟩ x 2 k and ⟨ ⟩ x 3 k are the only 6 -submodules of x k 6 , and their intersection is ( ) for every ∈ τ G, and every nonzero ∈ x M τ τ . If M is left σ-faithful for all ∈ σ G, then it is called left faithful. .
is not null graph. Then, M is σ-faithful for some ∈ σ G if and only if the map . Therefore, ϕ σ is a graph isomorphism. For the converse, suppose that there exists ( ) , then the following assertions hold: is connected if and only if ( ) G M σ is connected.
(i) The "if" part is Theorem 3.1. For the "only if" part, assume ( ) G R M Γ , , is connected and let N σ K and σ be two distinct vertices in ( ) , and hence, Hence, we obtain a path connecting Since all elements of β Nσ contain N σ , β Nσ is a clique in . Thus, ∪ ∈ β N C N σ σ is finite, which yields C itself is finite. Therefore, ( ( )) < ∞ ω G M σ .
For the converse, suppose that ( ( )) < ∞ ω G M σ , and for each ( ) Then, M σ is a direct sum of two simple R e -modules if and only if M is a direct sum of two G-graded simple R-modules.
Proof. The proof follows directly from Theorems 2.2, 2.4, and Part (i) of Theorem 3.5.

Clearly, ( )
For the remainder of this section, we focus on the relationships between the graph-theoretic properties of ( ) G R M Γ , , and ( ) G M when the grading group is a linearly ordered group. For details on rings and modules graded by linearly ordered group, see [21,Chapter 5].
A linearly ordered group is a group G equipped with a total ordered relation ≤ such that for all ∈ α β δ G , , , ≤ α β implies ≤ αδ βδ and ≤ δα δβ.
is well ordered subset of G and ⊆ N K , then = N K if and only if = N KT is connected if and only if ( ) G M is connected.
is connected, then ( ) G M is not null graph, and therefore, it is connected. For the converse, assume that ( ) G M is connected and let N and K be adjacent vertices of ( ) is not null, and hence, it is connected. □ Proof. Clearly, if ( ( )) = ∞ g G M , then ( ( )) = ∞ g G R M Γ , , . Assume that ( ( )) < ∞ g G M , it follows from [11, Theorem 2.5] that ( ( )) = g G M 3. If M is not Noetherian, then there are three nontrivial R-submodules N 1 , N 2 , and N 3 such that ⊊ ⊊ N N N 1 2 3 . Then, by part (iv) of Lemma 3.11, we obtain that ⊊ ⊊ N N Ñ~1 . Again by part (iv) of Lemma 3.11, we obtain the 3-cycle ( . Therefore, ( ( )) = g G R M Γ , , 3. □ The length of an R-module M over R, denoted by ( ) ℓ M , is the supremum of the lengths of chains of R-submodules of M.

Proof.
(i) Follows directly from the fact that ( ) G R M Γ , , is a subgraph of ( ) G M . (ii) Since ( ( )) = ∞ g G R M Γ , , , it follows from Theorem 2.11 that M is G-graded local, and since ( ( )) ( ( )) ≠ g G M g G R M Γ , , , by Theorem 3.13, M is not local. (iii) Since M is not local, ( ) ℓ ≥ M 3. Assume there are three R-submodules N 1 , N 2 , and N 3 such that ⊊ ⊊ N N N 1 2 3 . Then, by part (iv) of Lemma 3.11, we obtain that ⊊ ⊊ N N Ñ~1 □