Intersection graphs of graded ideals of graded rings

: In this article, we introduce and study the intersection graph of graded ideals of a graded ring. The intersection graph of G − graded ideals of a graded ring R , denoted by Gr G ( R ), is undirected simple graph defined on the set of nontrivial graded left ideals of R , such that two left ideals are adjacent if their intersection is not trivial. We study properties for these graphs such as connectivity, regularity, completeness, domination numbers, and girth. We also present several results on the intersection graphs related to faithful grading, strong grading, and graded idealization.


Introduction
Throughout this article, all rings are associative with unity 1 0. Let G be a multiplicative group with identity e. 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 στ for all σ, τ ∈ G. When R is G−graded we denote that by (R, G). The support of (R, G) is defined as supp(R, G) = {σ ∈ G : R σ 0}. The elements of R σ are called homogeneous of degree σ. The set of all homogeneous elements is denoted by h(R). If x ∈ R, then x can be written uniquely as σ∈G x σ , where x σ is the component of x in R σ . It is well known that R e is a subring of R with 1 ∈ R e . A left ideal I of R is called G−graded left ideal provided that I = ⊕ σ∈G (I∩R σ ).
In the last two decades, the theory of graded rings and modules has been receiving an increasing interest. Many authors introduced and studied, in a parallel way, the graded version of a wide range of concepts see [2, 10, 15-17, 20, 22, 23, 28, 29, 32]. Another area of research that developed remarkably in recent years is studying graphs associated to algebraic structures. These studies usually aim to investigate ring properties using graph theory concepts. Since Beck [11] introduced the concept of zero divisor graph in 1988, this approach became very popular. Other interesting examples of graphs associated to rings are total graphs, annihilating-ideal graph, and unit graphs (see [7,9,12,17]). For studies on graphs associated with graded rings and graded modules, in particular, see [21,30].
In 2009, Chakrabarty et al. [14] introduced the intersection graph of ideals of a ring. Denote by I * (R) the family of all nontrivial left ideals of a ring R. The intersection graph of ideals of R, denoted by G(R), is the simple graph whose set of vertices is I * (R), such that two vertices I and J are adjacent if I ∩ J {0}. Chakrabarty et al. [14] studied the connectivity of G(R) and investigated several properties of G(Z n ). Akbari et al. [5] studied these graphs more deeply. Among many results, they characterize all rings R for which G(R) is disconnected. For other interesting studies of intersection graphs of ideals of rings the reader is referred to [3, 4, 6, 18, 19, 25-27, 31, 33].
The main theme of this work is the study of a graded version of the intersection graph of left ideals. We introduce the intersection graph of the G−graded left ideals of a G−graded ring R denoted by Gr G (R). Definition 1.1. Let R be a G−graded ring. The intersection graph of the G−graded left ideals of R, denoted by Gr G (R), is the simple graph whose set of vertices consists of all nontrivial G−graded left ideals of R, such that two vertices I and J are adjacent only if I ∩ J {0}.
Sections 2 and 3 focus on the graph theory properties of Gr G (R). In particular, we discuss connectivity, diameter, regularity, completeness, domination numbers, and girth. Among many results, Theorem 2.6 gives necessary and sufficient conditions for the disconnectivity of Gr G (R). In Theorem 2.13, we describe the regularity of Gr G (R), and Theorem 3.5 classifies all gradings (R, G) for which g(Gr G (R)) = ∞. Many of these results are analogue to the nongraded case. Section 4 is devoted to the relationship between Gr G (R) and G(R e ) when the grading is faithful, strong, or first strong. In case of left e−faithful, we obtain an equivalence relation ≈ on vertices Gr G (R) by I ≈ J if and only if I ∩R e = J ∩R e . Then we are able to show that the quotient graph of Gr G (R) over the equivalence classes of ≈ is isomorphic to G(R e ). This isomorphism allows us to extent many of the graphical properties of G(R e ) to Gr G (R). Concerning strong grading, we prove that if (R, G) is first strong grading then Gr G (R) G(R e ). In this section also we study the the relationship between Gr G (R) and G(R) when the grading group G is an ordered group. The last section is devoted to the intersection graph of graded ideals of Z 2 −graded idealizations.
For standard terminology and notion in graph theory, we refer the reader to the text-book [13]. Let Γ be a simple graph with vertex set V(Γ) and set of edges E(Γ). Then |V(Γ)| is the order of Γ. If x, y ∈ V(Γ) are adjacent we write 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 (res. not adjacent). A complete (resp. null) graph with n vertices is denoted by K n (resp. N n ). A graph is called start graph if it has no cycles and has one vertex (the center) that is adjacent to all other vertices. A graph is said to be connected if any pair of its vertices is connected by a path. For any pair of vertices x, y in Γ, the distance d(x, y) is the length of the shortest path between them and diam(Γ) is the supremum of {d(x, y) | x, y ∈ V(Γ)}. The girth of a Γ, denoted by g(Γ) is the length of its shortest cycle. If Γ has no cycles then g(Γ) = ∞. A graph Υ is a subgraph of Γ if V(Υ) ⊆ V(Γ) and E(Υ) ⊆ E(Γ). Υ is called induced subgraph if any edge in Γ that joins two vertices in Υ is in Υ. A complete subgraph of Γ is called a clique, and the order of the largest clique in Γ, denoted by ω(Γ), is the clique number of Γ. A dominating set in Γ is a subset D of V(Γ) such that 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.

Connectivity, regularity and diameter of Gr G (R)
Let R be a G−graded ring. Denote by hI * (R) the set of all nontrivial G−graded left ideals of R.
{0} for all J ∈ I * (R) (resp. J ∈ hI * (R)). We call R G−graded left Noetherian (resp. Artinian) if R satisfies the ascending (resp. descending) chain condition for the G−graded left ideals. Analogously, we say R is G−graded local if it has a unique G−graded maximal left ideal. The ring R is called G−graded domain if it is commutative and has no homogeneous nonzero zero-divisors. Similarly, we call R a G−graded division ring if every nonzero homogeneous element is a unit. A G−graded field is a commutative G−graded division ring. Next we state a well known lemma regarding graded ideals, which will be used frequently throughout the paper. The following lemma is straightforward so we omit the proof.

I is G−graded essential if and only if N(I)
The following is a well known results about Z−graded fields (see [32]). Theorem 2.3. Let R be a commutative Z−graded ring. Then R is a Z−graded field if and only if R 0 is a field and either R = R 0 with trivial grading or Theorem 2.6 gives a necessary and sufficient condition for the intersection graph of graded ideals to be disconnected. We will see that this result is analogue to the nongraded case . First we state the result in nongraded case. Proof. Suppose that Gr G (R) is disconnected. For a contradiction, assume I and J are two adjacent vertices. So I, J, and I ∩ J belong to the same component of Gr G (R). Since Gr G (R) is disconnected, there is a vertex K that is not connected to anyone of the vertices I, J, and I ∩ J. If (I ∩ J) + K R then (I ∩ J) ∼ ((I ∩ J) + K) ∼ K is a path connecting I ∩ J and K, a contradiction. So (I ∩ J) + K = R. Now let a ∈ I. Then a = t + c for some t ∈ I ∩ J and c ∈ K. So a − t = c ∈ I ∩ K = {0}, consequently a = t ∈ I ∩ J. This implies that I = I ∩ J. Similarly, we get J = I ∩ J. Hence we have I = J a contradiction. Therefore Gr G (R) contains no edges, and hence it is a null graph. □ The following result is a direct consequence of Theorem 2.6.
Corollary 2.7. Let R be a G−graded ring. If Gr G (R) is disconnected then R contains at least two G−graded minimal left ideals and every G−graded left ideal of R is principal, graded minimal, and graded maximal.
It is known that if R 1 and R 2 are G−graded rings then Theorem 2.8. Let R be a commutative G−graded ring. Then Gr G (R) is disconnected if and only if R R 1 × R 2 where R 1 and R 2 are G−graded fields.
Proof. Assume Gr G (R) is disconnected. Then by Theorem 2.6 and Corollary 2.7, R has two G−graded maximal as well as G−graded minimal ideals I and J such that I + J = R and I ∩ J = {0}. Hence R/I and R/J are G−graded fields and R R/I × R/J. For the converse, assume that R R 1 × R 2 where R 1 and R 2 are G−graded fields. Then R 1 × 0 and 0 × R 2 are the only G−graded ideals of R. Hence Gr G (R) is disconnected. □ Corollary 2.9. Let R be a commutative G−graded ring. If Gr G (R) is connected, then every pair of G−graded maximal left ideals have non-trivial intersection.
Let R be a G−graded ring with at least two distinct nontrivial G−graded ideals. Since Gr G (R) is a subgraph of G(R), it follows that if Gr G (R) is connected then so is G(R). However, the converse of this statement need not be true. Indeed, Take a field K and let However, R 1 and R 2 are not fields, and hence G(R) is connected. In fact, from Theorem 2.3, Theorem 2.5, and Corollary 2.8 we get the following result.
Corollary 2.10. Let R be a commutative Z−graded ring such that Gr Z (R) is disconnected. Then R R 1 × R 2 such that one of the following is true: 1. R 1 and R 2 are fields, and hence G(R) is disconnected. Proof. Suppose R is G−graded domain. Then clearly R is G−graded reduced. Now, let I, J ∈ hI * (R), and take 0 a ∈ I ∩ h(R) and 0 b ∈ J ∩ h(R). Then 0 ab ∈ I ∩ J, and hence I and J are adjacent. Therefore Gr G (R) is complete. Conversely, suppose that R is G−graded reduced and Gr G (R) is complete. Assume that there are a, b ∈ h(R) \ {0} such that ab = 0. Since Gr G (R) is complete, there exists 0 c ∈ ⟨a⟩ ∩ ⟨b⟩ ∩ h(R). Hence c 2 ∈ ⟨a⟩ ⟨b⟩ = {0}. This implies that c 2 = 0, a contradiction. Therefore R is G−graded domain. □ Theorem 2.13. If R is a left G−graded Artinian ring such that Gr G (R) is not null graph, then the followings are equivalent: 1. Gr G (R) is regular.
2. R contains a unique G−graded minimal left ideal.

Domination, clique and girth of Gr G (R)
A commutative G−graded ring R is called G−graded decomposable if there is a pair of nontrivial G−graded ideals S and T of R, such that R S × T . If R is not G−graded decomposable then it is called G−graded indecomposable. Theorem 3.1. Let R be commutative G−graded ring. Then γ(Gr G (R)) ≤ 2. Furthermore the followings are true.

If R
S × T for some nontrivial graded ideals S , T of R then γ(Gr G (R)) = 2 if and only if γ(Gr G (S )) = γ(Gr G (T )) = 2. Proof. Let I 1 ⊇ I 2 ⊇ · · · I n · · · be a descending chain of G−graded left ideals. Then {I k } ∞ k=1 is a clique in Gr G (R), and hence it is finite. □ Proof. Assume gr(Gr G (R)) is finite and let I 0 ∼ I 1 ∼ · · · ∼ I n be a cycle.
The remaining case is that I 0 ∩ I 1 I 0 or I 1 . In this case we obtain the 3−cycle I 0 ∼ I 1 ∼ (I 0 ∩ I 1 ). Hence gr(Gr G (R)) = 3. □ The next theorem give a characterization of G−graded rings R such that g(Gr G (R)) = ∞. In fact, this result can be refer to as the graded version of [5,Theorem 17].
Theorem 3.5. Let R be a G−graded ring such that Gr G (R) is not a null graph. If gr(Gr G (R)) = ∞ then R is a G−graded local ring and Gr G (R) is a star whose center is the unique G−graded maximal left ideal of R, say M. Moreover, one of the followings hold: 1. M is principal. In this case Gr G (R) = K 1 or K 2 . {0} for all J ∈ hI * (R). Moreover, since Gr G (R) has no cycles then J ⊆ M for all J ∈ hI * (R). So fare we proved that Gr G (R) is a star whose center is M. Now we proceed to prove parts (1) and (2). Since R is left G−graded Artinian, by [24, Corollary 2.9.7] R is left G−graded Noetherian. So M is generated by a finite set of homogeneous elements. If a minimal set of homogeneous generators has at least three elements, containing say a, b, c, ..., then M ∼ (Ra + Rb) ∼ (Rb + Rc) is a 3−cycle in Gr G (R), a contradiction. So a minimal set of homogeneous generators of M has at most two elements. Moreover, since M is finitely generated and J g (R) = M (where J g (R) is the graded Jacobson radical of R, which is the intersection of all G−graded maximal left ideals), by [24, Corollary 2.9.2], M ⊋ M 2 ⊋ M 3 ⊋ · · · . In addition, since Gr G (R) has no 3−cycles, we get M 3 = 0. Case 1: Suppose M = Ra for some a ∈ h(R). Let I ∈ hI * (R) and let x ∈ I ∩ h(R). Then x = ya for some y ∈ R. Since x, a ∈ h(R), it results that y ∈ h(R). If y M, then Ry = R, because M is the only G−graded maximal left ideal. So y is a unit, and hence I = M. Assume y ∈ M. Then, we get x = wa 2 for some w ∈ h(R). If w ∈ M, then x ∈ Ra 3 = {0}, a contradiction. So w M, and hence I = Ra 2 . Therefore we have that if Ra 2 = 0 then Gr G (R) = K 1 , otherwise Gr G (R) = K 2 . Case 2: Assume the minimal set of homogeneous generators of M has two elements say a, b, consequently M = Ra + Rb. Since Gr G (R) has no 3−cycles, Ra and Rb are G−graded minimal. Moreover, we have Ra and Rb are left subideals of J g (R). By [24, Corollary 2.9.2] it results that (Ra) 2 = RaRb = RbRa = (Rb) 2 = 0, and hence M 2 = 0. □

Intersections graph of types of gradings
In this section we focus on the relationship between G(R e ) and Gr G (R). Note that if I e is left ideal of R e then RI e is a G−graded left ideal of R. Moreover, RI e ∩ R e = I e .
Theorem 4.1. Let R be a G−graded ring such that R e contains at least two nontrivial left ideals. If G(R e ) is connected then Gr G (R) is connected, and hence G(R) is connected.
Proof. Since G(R e ) is connected then it must contain an edge. Let I e , J e be two adjacent vertices of G(R e ). Then RI e and RJ e are vertices in Gr G (R). Moreover RI e ∩ R e = I e and RJ e ∩ R e = J e , and so RI e RJ e . Additionally, we have {0} I e ∩ J e ⊆ RI e ∩ RJ e . Therefore Gr G (R) is not null, and hence it is connected. □ The converse of Theorem 4.1 need not be true. Indeed, let R e = Z pq , where p and q are distinct primes, and Take R = R e [x] with Z−grading R k = R e x k , k ≥ 0 and R k = 0, k < 0. The ideals Rx and Rx 2 are adjacent in Gr G (R) and so Gr G (R) is connected, while G(R e ) is disconnected because it has two minimal ideals.
A grading (R, G) is called left σ−faithful for some σ ∈ G, if R στ −1 x τ {0} for every τ ∈ G, and every nonzero x τ ∈ R τ . If (R, G) is left σ−faithful for all σ ∈ G then it is called left faithful. Proof. Suppose (R, G) is left σ−faithful for some σ ∈ G. Let I ∈ hI * (R) and take a nonzero element . If x τ is a nonzero homogenous element of degree τ, for some τ ∈ G, then Rx τ ∈ hI * (R). So by assumption,    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 [1,23,29]). It is know that (R, G) is strong if and only if R τ R σ = R τσ for all τ, σ ∈ G. In [23,Corollary 1.4] it is proven that if (R, G) is a strong grading and I is a left G−graded ideal of R, then I = RI e , where I e = I ∩ R e . In fact this result is still true in case H = supp(R, G) is a subgroup of G and R = ⊕ σ∈H R σ is a strongly H−graded ring. Fact 2.5 in [29] states that (R, G) is first strong if and only if H = supp(R, G) ≤ G and (R, H) is strong. So next we state a weaker version of [23,Corollary 1.4].
Lemma 4.8. Let (R, G) be first strong grading. Then for every I ∈ hI * (R), I = RI e , where I e = I ∩ R e . Theorem 4.9. Let (R, G) be first strong grading. Then G(R e ) Gr G (R).
Proof. Since (R, G) is first strong , so by Lemma 4.8, we have hI * (R) = {RI e | I e ∈ I * (R e )}. Moreover (R, G) is left e-faithful, because if for some τ ∈ supp(R, G) and x τ ∈ R τ , we have R τ −1 x τ = {0}, then R e x τ = R τ R τ −1 x τ = {0}, and hence x τ = 0. Now the result follows by Theorem 4.3. □ The rest of this section is devoted to study the relationship between Gr G (R) and G(R) when the grading group is an ordered group. An ordered group is a group G together with a subset S such that

Conclusions
In this study, we introduced the notions of intersection graph of graded left ideals of graded rings, namely Gr G (R). Several properties of these graphs such as connectivity, regularity, completeness, and girth have been discussed. In addition, we investigated the relationship between Gr G (R) and the intersection graph of left ideals of the identity component, G(R e ), when the grading is faithful, strong, or first strong. We also studied the relationship between Gr G (R) and G(R) when the grading group is an ordered group. As a proposal of further work, one may study the graded case of other types of graphs associated to rings such as zero-divisor graphs, annihilating-ideal graph, and unit graphs.