From the journal Open Mathematics

## Abstract

MSC 2010: 13A02; 05C25; 16W50

## 1 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 G be a group. A ring R is said to be G -graded if there exist additive subgroups { R σ σ G } such that R = σ G R σ and R σ R τ R σ τ for all σ , τ G . A left R -module M is said to be G -graded if there exist additive subgroups M σ of M indexed by the elements σ G such that M = σ G M σ and R τ M σ M τ σ for all τ , σ G . The elements of M σ are called homogeneous of degree σ . If x M , then x can be written uniquely as σ G x σ , where x σ is the component of x in M σ . An R -submodule N of M is called G -graded provided that N = σ G ( N M σ ) . We denote by h S ( M ) the set of all nontrivial G -graded R -submodules of M .

## Definition 1.1

Let R be a G -graded ring and M be a G -graded left R -module. The intersection graph of G -graded submodules of M , denoted by Γ ( G , R , M ) , is defined to be an undirected simple graph whose set of vertices is h S ( M ) and two vertices N and K are adjacent if N K { 0 } .

We aim to study the properties of these graphs analogous to the nongraded case. In addition, we investigate connections and relationships among G ( M σ ) , Γ ( G , R , M ) , 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 Γ ( G , R , M ) . We discuss their connectivity, diameter, regularity, completeness, domination numbers, and girth. In Section 3, we investigate the relationships between Γ ( G , R , M ) 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 Γ ( G , R , M ) 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 V ( Γ ) and E ( Γ ) , respectively. The cardinality V ( Γ ) is referred to as the order of Γ . If x , y V ( Γ ) are adjacent, we denote that as x y . The neighborhood of a vertex x is N ( x ) = { y V ( Γ ) y x } , and the degree of x is deg ( x ) = N ( x ) . 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.

## 2 Graph theoretic properties of Γ ( G , R , M )

We present the following well-known technical lemma in this section.

## Lemma 2.1

[20, Lemma 2.1] Let R be a G -graded ring and M be a G -graded R -module.

1. If I and J are G -graded ideals of R , then I + J and I J are G -graded ideals of R .

2. If N and K are G -graded R -submodules of M , then N + K and N K are G -graded R -submodules of M .

3. If N is a G -graded R -submodule of M , r h ( R ) , x h ( M ) and I is a G -graded ideal of R , then R x , I N , and r N are G -graded R -submodules of M . Moreover, ( N : R M ) = { r R : r M N } is a G -graded ideal of R .

The following two results from [8] classify disconnected intersection graphs of submodules.

## Theorem 2.2

[8, Theorem 2.1] Let M be an R -module. Then, the graph G ( M ) is disconnected if and only if M is a direct sum of two simple R -modules.

## Corollary 2.3

[8, Corollary 2.3] Let M be an R -module. Then, the graph G ( M ) is disconnected if and only if it is null graph with at least two vertices.

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 Γ ( G , R , M ) 2 . Then, the followings are equivalent:

1. Γ ( G , R , M ) is disconnected.

2. Γ ( G , R , M ) is a null graph.

3. Every nontrivial G -graded R -submodule of M is G -graded maximal as well as G -graded simple.

4. M is a direct sum of two G -graded simple (or maximal) R -modules.

## Proof

( 1 ) ( 2 ) Suppose that Γ ( G , R , M ) 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 Γ ( G , R , M ) . Since Γ ( G , R , M ) is disconnected, there is a vertex L that is not connected to any of the vertices N , K , and N K . If ( N K ) + L M , then ( N K ) ( ( N K ) + L ) L is a path connecting N K and L , a contradiction. So ( N K ) + L = M . Now let a N . Then, a = t + c for some t N K and c L . So a t = c N L = { 0 } ; consequently, a = t N K . This implies that N = N K . Similarly, we obtain K = N K . Hence, we have N = K , a contradiction. Therefore, Γ ( G , R , M ) contains no edges, and hence, it is a null graph.

( 2 ) ( 3 ) Straightforward.

( 3 ) ( 4 ) Let N and K be G -graded maximal as well as G -graded simple R -submodules of M . Then, N + K = M and N K = { 0 } . Hence, N and K are G -graded simple R -modules and M = N K .

( 4 ) ( 1 ) Suppose M = N K , where N and K are G -graded simple R -modules. Then, N and K are G -graded simple R -submodules. Also, they are G -graded maximal because N M K and K M N . Thus, N and K are isolated vertices. Therefore, Γ ( G , R , M ) is disconnected.□

Roshan-Shekalgourabi and Hassanzadeh-Lelekaami [6] associated a graph G M with a G -graded R -module M , where V ( G M ) = h S ( M ) and two nontrivial G -graded R -submodules N and K are adjacent if N + K = M . Clearly, the two concepts G M and Γ ( G , R , M ) are distinct. The next theorem presents an obvious relation between these two graphs.

## Corollary 2.5

Let M be a G -graded R -module. Then, Γ ( G , R , M ) 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.

## Corollary 2.6

Let M be a G -graded R -module. If Γ ( G , R , M ) is connected, then every pair of G -graded maximal R -submodules intersect nontrivially.

The distance d ( x , y ) between any two vertices x , y in a graph Γ is the length of the shortest path between them, and diam ( Γ ) is the supremum of { d ( x , y ) x , y V ( Γ ) } .

## Theorem 2.7

Let M be a G -graded R -module. If Γ ( G , R , M ) is connected, then diam ( Γ ( G , R , M ) ) 2 .

## Proof

Suppose N and K are distinct vertices in Γ ( G , R , M ) . If N and K are adjacent, then d ( N , K ) = 1 . If N and K are nonadjacent, then d ( N , K ) 2 . If N K M , then we have the path N N K K , and hence, d ( N , K ) = 2 . If N K = M , then either N or K is not G -graded simple, say N . Let ( 0 ) L N . Thus, we have the path N L K K , and hence, d ( N , K ) = 2 . As a result, d ( N , K ) 2 .□

## Theorem 2.8

Let M be a G -graded Artinian R -module such that Γ ( G , R , M ) is not a null graph. Then, the followings are equivalent:

1. Γ ( G , R , M ) is regular.

2. GMin ( M ) = 1 .

3. Γ ( G , R , M ) is complete.

## Proof

( i ) ( i i ) Suppose Γ ( G , R , M ) 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 N Y . This implies that N ( N ) N ( Y ) ; consequently, deg ( Y ) > deg ( N ) , a contradiction. Hence, M contains a unique G -graded simple R -submodule.

( i i ) ( i i i ) Suppose M contains a unique G -graded simple R -submodule, say N . Since M is G -graded Artinian, N K for all K h S ( M ) . Thus, Γ ( G , R , M ) is complete.

( i i i ) ( i ) Straightforward.□

## Remark 2.9

In a G -graded R -module M , a G -graded submodule N of M is called G -graded essential if N K ( 0 ) for all K h S ( M ) . The graded socle, Gsoc ( M ) , of M is the sum of all G -graded simple R -submodules of M . Equivalently Gsoc ( M ) equals the intersection of all G -graded essential R -submodules of M , see [21, page 48]. So, if M is G -graded Artinian and Γ ( G , R , M ) is complete, then every G -graded R -submodule is G -essential, and thus, by Theorem 2.8, GMin ( M ) = Gsoc ( M ) .

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 ) = .

## Theorem 2.10

If M is a G -graded R -module, then g r ( Γ ( G , R , M ) ) { 3 , } .

## Proof

Assume g ( Γ ( G , R , M ) ) < and g ( Γ ( G , R , M ) ) 4 . This implies that every pair of distinct nontrivial G -graded submodules of M with nonzero intersection should be comparable, otherwise Γ ( G , R , M ) will have a cycle of length 3, a contradiction. Since g ( Γ ( G , R , M ) ) 4 , Γ ( G , R , M ) 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 Γ ( G , R , M ) , 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, L P ( 0 ) . Thus, L K P L is a cycle of length 3 in Γ ( G , R , M ) , a contradiction. In the second case, we have ( 0 ) L N K , and therefore, N L K N is a cycle of length 3 in Γ ( G , R , M ) , which again yields a contradiction. Therefore, g ( Γ ( G , R , M ) ) = 3 .□

The next theorem gives a characterization of G -graded R -modules such that g ( Γ ( G , R , M ) ) = . 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.

## Theorem 2.11

Let M be a G -graded Noetherian R -module such that Γ ( G , R , M ) is not a null graph with Γ ( G , R , M ) 2 . If g ( Γ ( G , R , M ) ) = , then M is a G -graded local module and Γ ( G , R , M ) is a star graph whose center is the unique G -graded maximal R -submodule of M .

## Proof

By Theorem 2.4, Γ ( G , R , M ) is connected. Suppose that N 1 and N 2 are two distinct G -graded maximal R -submodules of M . By Theorem 2.7, d ( N 1 , N 2 ) 2 . If N 1 N 2 ( 0 ) , then N 1 ( N 1 N 2 ) N 2 N 1 is a 3-cycle, a contradiction. So N 1 N 2 = ( 0 ) . Since N 1 and N 2 are G -graded maximal R -submodules, we obtain M = N 1 N 2 . Thus, Γ ( G , R , M ) is null, which contradicts the assumption that Γ ( G , R , M ) 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 N K = K ( 0 ) for all K h S ( M ) . However, since Γ ( G , R , M ) has no cycles, we conclude that Γ ( G , R , M ) 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 .

## Lemma 2.12

Let M be a G -graded R -module. If ω ( Γ ( G , R , M ) ) < , 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 Γ ( G , R , M ) , and hence, the chain is finite.□

## Corollary 2.13

Let M be a G -graded R -module. If Γ ( G , R , M ) is connected, then GMax ( M ) ω ( Γ ( G , R , M ) ) .

## Proof

If Γ ( G , R , M ) is connected, then by Corollary 2.6, GMax ( M ) is a clique, and thus, GMax ( M ) ω ( Γ ( G , R , M ) ) .□

A subset D of the set of vertices of a graph Γ is called a dominating set in Γ if every vertex of Γ is in D or adjacent to a vertex in D . The domination number of Γ , denoted by γ ( Γ ) , is the minimum cardinality of a dominating set in Γ . In the next theorem, we determine the domination number of Γ ( G , R , M ) . 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, γ ( Γ ( G , R , M ) ) 2 . Furthermore, if M is G -graded indecomposable, then γ ( Γ ( G , R , M ) ) = 1 .

## Proof

Let N be a G -graded maximal R -submodule of M . If there exists K h S ( M ) such that N K = ( 0 ) , then N + K = M , and hence, M = N K . So the set { N , K } is a dominating set, and thus, γ ( Γ ( G , R , M ) ) 2 . This proves the first part.

If M is G -graded indecomposable, then N K ( 0 ) for all K h S ( M ) . Consequently, { N } is a dominating set, and hence, γ ( Γ ( G , R , M ) ) = 1 .□

## 3 Intersection graph of types of gradings

In this section, we study some relationships between Γ ( G , R , M ) and G ( M σ ) and between Γ ( G , R , M ) and G ( M ) . It is well known that if M is a G -graded R -module, then M σ is an R e -module for each σ G . So G ( M σ ) here represents the intersection graph of R e -submodules of M σ . We also note that if N σ is an R e -submodule of M σ , then R N σ is a G -graded R -submodule of M and R N σ M σ = N σ .

## Theorem 3.1

Let M be a G -graded R -module. If for some σ G , G ( M σ ) is connected with at least two vertices, then Γ ( G , R , M ) is connected, and hence, G ( M ) is connected.

## Proof

Since G ( M σ ) is connected, it must contain an edge. Let N σ , K σ be two adjacent vertices in G ( M σ ) . Then, R N σ and R K σ are vertices in Γ ( G , R , M ) . Moreover, R N σ M σ = N σ and R K σ M σ = K σ , and so, R N σ R K σ . In addition, we have { 0 } N σ K σ R N σ R K σ . Therefore, Γ ( G , R , M ) is not null, and hence, it is connected. The last part follows from Corollary 2.3 because Γ ( G , R , M ) is a subgraph of G ( M ) .□

## Remark 3.2

The converse of Theorem 3.1 needs not to be true in general. Let R = Z 6 with trivial Z -grading; that is, R 0 = Z 6 , and R k = 0 , for all k 0 , and choose M = Z 6 [ x ] as Z 6 -module with grading M k = Z 6 x k , k 0 , and M k = 0 , k < 0 . The Z -graded Z 6 -submodules Z 6 and Z 6 + Z 6 x are adjacent in Γ ( Z , Z 6 , Z 6 [ x ] ) , and by Theorem 2.4, we have Γ ( Z , Z 6 , Z 6 [ x ] ) is connected. On the other hand, for each k 0 , 2 x k and 3 x k are the only Z 6 -submodules of Z 6 x k , and their intersection is ( 0 ) . So G ( Z 6 x k ) is disconnected for all k 0 .

A G -graded R -module M is said to be left σ -faithful for some σ G , if R σ τ 1 x τ { 0 } for every τ G , and every nonzero x τ M τ . If M is left σ -faithful for all σ G , then it is called left faithful.

## Lemma 3.3

[21, Proposition 2.6.3] A G -graded R -module M is σ -faithful for some σ G if and only if N M σ { 0 } for all N h S ( M ) .

Let M be a G -graded R -module. Define the simple graph Γ σ ( G , R , M ) on h S ( M ) , where N and K are adjacent only if N K M σ { 0 } . We will call this graph the σ -intersection graph of G -graded R -modules of M . It is clear that if N K M σ { 0 } , then N K { 0 } . So Γ σ ( G , R , M ) is a subgraph of Γ ( G , R , M ) .

## Theorem 3.4

Let M be a G -graded R -module such that Γ ( G , R , M ) is not null graph. Then, M is σ -faithful for some σ G if and only if the map ϕ σ : Γ σ ( G , R , M ) Γ ( G , R , M ) defined by ϕ ( N ) = N is a graph isomorphism.

## Proof

Suppose M is σ -faithful for some σ G . Clearly, ϕ is a set bijection. Let N , K h S ( M ) such that N K { 0 } . Since M is σ -faithful, by Lemma 3.3, N K M σ { 0 } , which implies that N and K are adjacent in Γ σ ( G , R , M ) . Therefore, ϕ σ is a graph isomorphism. For the converse, suppose that there exists N h S ( M ) such that N M σ = { 0 } . Then, N is an isolated vertex in Γ σ ( G , R , M ) , which implies that N is an isolated vertex in Γ ( G , R , M ) because ϕ σ is an isomorphism. So Γ ( G , R , M ) is null, a contradiction. Then, N M σ { 0 } for all N h S ( M ) . Hence, by Lemma 3.3, M is σ -faithful.□

## Theorem 3.5

Let M be a σ -faithful G -graded R -module. If R M σ = M , then the following assertions hold:

1. Γ ( G , R , M ) is connected if and only if G ( M σ ) is connected.

2. γ ( Γ ( G , R , M ) ) = γ ( G ( M σ ) ) .

3. ω ( Γ ( G , R , M ) ) < if and only if ω ( G ( M σ ) ) < , and for each N σ S ( M σ ) , the set β N σ = { N h S ( M ) N M σ = N σ } is finite.

## Proof

(i) The “if” part is Theorem 3.1. For the “only if” part, assume Γ ( G , R , M ) is connected and let N σ and K σ be two distinct vertices in G ( M σ ) . If R N σ R K σ { 0 } , then by Theorem 3.4, R N σ R K σ M σ { 0 } . So we have N σ K σ = R N σ M σ R K σ M σ = R N σ R K σ M σ { 0 } , and hence, N σ K σ is a path. Assume R N σ R K σ = { 0 } . By Theorem 2.7, there is Y h S ( M ) such that R N σ Y { 0 } and R K σ Y { 0 } . Then, R N σ Y M σ { 0 } and R K σ Y M σ { 0 } . Since ϕ σ is a graph isomorphism, N σ ( Y M σ ) and K σ ( Y M σ ) are nontrivial. Moreover, Y M σ M σ because R M σ = M . Hence, we obtain a path connecting N σ and K σ in G ( M ) . Therefore, G ( M ) is connected.

(ii) Let S S ( M σ ) be a minimal dominating set in G ( M σ ) , and let S = { R N σ N σ S } . Clearly, S = S . Let K h S ( M ) such that K S . By Lemma 3.3, we have K M σ { 0 } , and hence, K M σ N σ { 0 } for some N σ S . So we have R N σ S and K is adjacent to R N σ in Γ ( G , R , M ) . Hence, S is a dominating set in Γ ( G , R , M ) . Therefore, γ ( Γ ( G , R , M ) ) γ ( G ( M σ ) ) . Now assume S is a minimal dominating set in Γ ( G , R , M ) , and let S = { N M σ N S } . Let K σ S ( M σ ) such that K σ S . If R K σ S , then R K σ M σ S . We have K σ ( R K σ M σ ) = K σ ( 0 ) . So K σ is adjacent to R K σ M σ S . Now assume R K σ S . So there exists N S such that R K σ N ( 0 ) . Hence, by Theorem 3.4, we obtain ( 0 ) R K σ N M σ K σ ( N M σ ) . Thus, K σ ( N M σ ) ( 0 ) , and so S is a dominating set in G ( M σ ) . So γ ( G ( M σ ) ) S S = γ ( Γ ( G , R , M ) ) .

(iii) Suppose ω ( Γ ( G , R , M ) ) < . Let N σ S ( M σ ) . Since all elements of β N σ contain N σ , β N σ is a clique in Γ ( G , R , M ) . Hence, β N σ ω ( Γ ( G , R , M ) ) < . Let C be a clique in G ( M σ ) . Then, N σ C β N σ is a clique in Γ ( G , R , M ) . Thus, N σ C β N 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 N σ S ( M σ ) , the set β N σ is finite. Let D be a clique in Γ ( G , R , M ) . Let Λ be the set of all N σ S ( M σ ) such that D β N σ is nonempty. Clearly, the collection { D β N σ N σ Λ } is a partition of D . So D = N σ Λ ( D β N σ ) . We note that, by the assumption for the converse, D β N σ is finite for all N σ Λ . Let N σ , K σ Λ . Then, there are N , K D such that N σ = N M σ and K σ = K M σ . Since D be a clique, N K ( 0 ) . In addition, because the grading is σ -faithful, by Lemma 3.3, we obtain that ( 0 ) ( N K ) M σ = N σ K σ . So Λ is a clique in G ( M σ ) , and hence, it is finite. This implies that D = N σ Λ ( D β N σ ) is finite. We just proved that every clique in Γ ( G , R , M ) is finite. Therefore, ω ( Γ ( G , R , M ) ) < .□

## Corollary 3.6

Let M be a σ -faithful G -graded R -module such that R M σ = M . 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.□

A grading ( R , G ) is called strong (resp. first strong) if 1 R σ R σ 1 for all σ G (resp. σ supp ( R , G ) ) (see [22,23]). In what follows, the symbol means “a subgroup of,” while the symbol means “isomorphic to.”

## Lemma 3.7

[23, Fact 2.5] A grading ( R , G ) is first strong if and only if H = supp ( R , G ) G and ( R , H ) is strong.

## Lemma 3.8

Let ( R , G ) be first strong grading and M be a G -graded R -module. If supp ( M , G ) supp ( R , G ) , then Γ ( G , R , M ) G ( M σ ) for all σ supp ( M , G ) .

## Proof

Fix σ supp ( M , G ) . We claim that if N is G -graded R -submodule of M , then N = R ( N M σ ) . Let 0 x N M τ for some τ G . Now σ , τ supp ( M , G ) supp ( R , G ) . Thus, since supp ( R , G ) G , τ σ 1 supp ( R , G ) . So R τ σ 1 R σ τ 1 = R e . This implies that 1 = i = 1 n r i s i for some r i R τ σ 1 and s i R σ τ 1 . Hence, x = i = 1 n r i s i x . Since x M τ and s i R σ τ 1 , s i x R σ τ 1 M τ M σ , for all i . Also s i x N because N is an R -submodule. So x R ( N M σ ) . Hence, N M τ R ( N M σ ) for all τ supp ( M , G ) . This implies that R ( N M σ ) N = τ G ( N M τ ) R ( N M σ ) . Hence, N = R ( N M σ ) . From the claim, we conclude that M = R ( M M σ ) = R M σ and N M σ { 0 } for all N h S ( M ) . Therefore, the correspondence N N M σ yields an isomorphism between Γ ( G , R , M ) and G ( M σ ) .□

## Corollary 3.9

Let ( R , G ) be strong grading and M be a G -graded R -module. Then, Γ ( G , R , M ) G ( M σ ) for all σ G .

## Example 3.10

Let A be a ring, and consider the ring R = M 3 ( A ) and the left R -module M = M 3 × 1 ( A ) with Z 2 -gradings given by

R 0 = A A 0 A A 0 0 0 A , R 1 = 0 0 A 0 0 A A A 0 . M 0 = A A 0 , M 1 = 0 0 A .

Clearly, ( R , Z 2 ) is strong. So by Corollary 3.9, G ( Z 2 , M 3 ( A ) , M 3 × 1 ( A ) ) G ( M 1 ) . The nontrivial R 0 -submodules of M 1 are given as follows:

0 0 I I I ( A ) .

Hence, G ( Z 2 , M 3 ( A ) , M 3 × 1 ( A ) ) G ( M 1 ) G ( A ) , where A is considered as left A -module.

For the remainder of this section, we focus on the relationships between the graph-theoretic properties of Γ ( G , R , M ) 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 α δ β δ and δ α δ β .

Suppose that M is G -graded R -module where G is a linearly ordered group. Then, any x M can be written uniquely as x = x σ 1 + x σ 2 + + x σ n , with σ 1 < σ 2 < < σ n . We call x σ n the homogeneous components of x of highest degree. For each R -submodule N of M , the G -graded R -submodule generated by the homogeneous components of the highest degrees of all elements of N is denoted by N ; that is, y is one of the generators of N if and only if there exists x = x σ 1 + x σ 2 + + x σ n N , with σ 1 < σ 2 < < σ n and x σ n = y . We have the following result from [21, Lemma 5.3.1, Corollary 5.3.3]

## Lemma 3.11

Let M be a G -graded R -module, where G is linearly ordered group and N and K are submodules of M . Then,

1. N = N if and only if N is G -graded R -submodule.

2. N = { 0 } if and only if N = { 0 } .

3. If N K , then N K .

4. If supp ( M , G ) is well ordered subset of G and N K , then N = K if and only if N = K

## Theorem 3.12

Let M be a G -graded R -module, where G is linearly ordered group. If supp ( M , G ) is well ordered subset of G , then Γ ( G , R , M ) is connected if and only if G ( M ) is connected.

## Proof

If Γ ( G , R , M ) 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 G ( M ) . Hence, N K { 0 } . Let J = N K . Since N K , either J N or J K . Without loss of generality, assume J N . Then, by parts (ii)–(iv) of Lemma 3.11, we have { 0 } J K . So Γ ( G , R , M ) is not null, and hence, it is connected.□

## Theorem 3.13

Let M be a G -graded R -module, where G is a linearly ordered group and supp ( M , G ) is well ordered subset of G . If M is local or not Noetherian, then g ( Γ ( G , R , M ) ) = g ( G ( M ) ) .

## Proof

Clearly, if g ( G ( M ) ) = , then g ( Γ ( G , R , M ) ) = . 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 1 N 2 N 3 . Then, by part (iv) of Lemma 3.11, we obtain that N 1 N 2 N 3 . Hence, N 1 N 2 N 3 N 1 is a 3-cycle in Γ ( G , R , M ) . Now suppose that M is Noetherian and local with maximal R -submodule N . This implies that K N for all K S ( M ) . Since G ( M ) is not a star graph, there are two distinct R -submodules K , L S ( M ) { N } such that K L { 0 } . Without the loss of generality, we may assume that K L K . So we have { 0 } K L K N . Again by part (iv) of Lemma 3.11, we obtain the 3-cycle ( K L ) K N ( K L ) in Γ ( G , R , M ) . Therefore, g ( Γ ( G , R , M ) ) = 3 .□

The length of an R -module M over R , denoted by ( M ) , is the supremum of the lengths of chains of R -submodules of M .

## Theorem 3.14

Let M be a G -graded R -module, where G is a linearly ordered group and supp ( R , G ) is well ordered subset of G . If g ( G ( M ) ) g ( Γ ( G , R , M ) ) , then the following assertions hold

1. g ( G ( M ) ) = 3 and g ( Γ ( G , R , M ) ) = .

2. M is G -graded local but not local.

3. The length of M over R is ( M ) = 3 .

4. For every maximal R -submodule K , K = N , where N is the unique G -graded maximal R -submodule of M .

5. If rad ( M ) = { 0 } , then ω ( G ( M ) ) = Max ( M ) . Otherwise, ω ( G ( M ) ) = Max ( M ) + 1 . ( rad ( M ) is the intersection of all maximal R -submodules of M .)

## Proof.

(i) Follows directly from the fact that Γ ( G , R , M ) is a subgraph of G ( M ) .

(ii) Since g ( Γ ( G , R , M ) ) = , it follows from Theorem 2.11 that M is G -graded local, and since g ( G ( M ) ) g ( Γ ( G , R , M ) ) , by Theorem 3.13, M is not local.

(iii) Since M is not local, ( M ) 3 . Assume there are three R -submodules N 1 , N 2 , and N 3 such that N 1 N 2 N 3 . Then, by part (iv) of Lemma 3.11, we obtain that N 1 N 2 N 3 ; consequently, g ( Γ ( G , R , M ) ) = 3 , a contradiction. So ( M ) = 3 .

(iv) Let K be a maximal R -submodule of M . Since G ( M ) is connected, K is not simple. So there is a nontrivial R -submodule L K . So L K . Also Γ ( G , R , M ) is a star graph, and hence, K = N .

(v) Since G ( M ) is connected, Max ( M ) is a clique in G ( M ) . Now from part (iii), we obtain that every R -submodule is either maximal or simple. Moreover, rad ( M ) Max ( M ) because M is not local. Therefore, if rad ( M ) = { 0 } , then Max ( M ) is clique of maximum size, otherwise Max ( M ) { rad ( M ) } is clique of maximum size.□

## Acknowledgements

The author thanks the referees for their valuable comments and constructive suggestions that helped improving this paper.

1. Conflict of interest: The author states no conflict of interest.

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