On a graph of ideals of a commutative ring

In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J). In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which G(R) is planar, complete and complete r-partite. We show that, if (R,M) is a local Artinian ring, then G(R) is complete if and only if Soc(R) is simple. Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.


Introduction
Over the last years, there has been an explosion of interest in associating a graph to an algebraic structure. In 1988, Istvan Beck proposed the study of commutative rings by representing them as graphs [6]. Since then a huge number of works have been added to the literature, see for instance [2,3,4,10,11,12,13,14,15,16,17,18]. Most properties of a ring are connected to a behavior of its ideals. Besides, ideals play crucial roles in the study of ring constructions. This is why it is interesting and useful to associate graphs to ideals of a ring, as for example in [1,10,24]. The bene…t of studying these graphs is that one may …nd some results about the algebraic structures.
A well known result of Ikeda and Nakayama [21] explores that if a (not necessarily commutative) ring R is right self-injective, then ann l (I \ J) = ann l (I) + ann l (J) for all right ideals I and J of R, where ann l (X) denotes the left annihilator of X. The study of rings that satisfying the aforementioned property has been initiated by Camillo, Nicholson and Yousif in [9] and called Ikeda-Nakayama rings.
In this paper, we introduce and investigate a new graph in a commutative ring R, in order to know when ann(I \ J) = ann(I) + ann(J) for ideals I and J of R.
Our main goal is to study the connection between the algebraic properties of a ring and the graph theoretic properties of the graph associated to it. We associate a graph G(R) to a commutative ring R whose vertices are nonzero proper ideals of R and two vertices I and J are adjacent if and only if ann(I \ J) = ann(I) + ann(J).
We summarize the contents of this article as follows. In section 2, we show that G(R) is a connected graph with diam(G(R)) 2 f0; 1; 2g and gr(G(R)) 2 f3; 1g. Also, we show that what happen for R and G(R), if gr(G(R)) = 1. In this section it is shown that G(R) is a star graph if and only if G(R) is a bipartite graph if and only if G(R) contains a cut vertex. In section 3, we investigate the planar property, complete or complete r-partite property of G(R). In this section maximal ideals and socle are useful instrument which help us to do our study. We prove that, if (R; M ) is a local Artinian ring, then the G(R) is complete if and only if Soc(R) is simple. It is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R=I) is complete for each ideal I of R. A complete characterizations of rings for which G(R) is planar or complete r-partite are provided. It is proved that, if R is a ring, then G(R) is planar if and only if one of the following holds: (1) R = F S, where F is a …eld and (S; M ) is a local ring with the only non-zero (2) (R; M ) is a local ring with the maximal ideal M and R is Ikeda-Nakayama with at most four nontrivial ideals.
(3) (R; M ) is a local ring with the maximal ideal M , M = Rx + Ry, M 2 = 0, all proper ideals, di¤erent from M , must be principal and of the form Rx, Ry, or R(x + ay), where a is an invertible element of R and G(R) is a star graph.
(4) (R; M ) is a local ring with the maximal ideal M , M = Rx + Ry, M 3 = 0 and the set of nontrivial ideals of R is equal to fM; Rx; Ry; Ry 2 ; R(x + y); R(x + y 2 ); Rx Ry 2 = Soc(R)g: Furthermore, it is shown that if R is a ring with G(R) is a complete r-partite graph with part ; s V i (1 i r), then R is Artinian and one of the following statements hold: (1) G(R) is complete and R is an Ikeda-Nakayama ring.
(2) R is a local ring,`(Soc(R)) 2 and if I; J 2 V i , then I and J are cyclic local R-modules with common maximal submodule I \ J.
Among other results, we give a description of a lower bound for the clique number of G(R).
In order to make this paper easier to follow, we recall in this section various notions which will be used in the sequel. Throughout this paper all rings are commutative with non-zero identity. Let R be a ring. By I(R), we denote the set of all nontrivial ideals of R. A ring R is said to be local if it has a unique maximal ideal M and we denote it by (R; M ). For a subset X of a ring R, the annihilator of X in R is ann(X) = fr 2 R : rx = 0 f or all x 2 Xg and we denote the set of all maximal ideals of R by max(R). The socle of ring R, denoted by Soc(R), is the sum of all minimal ideals of R. Following [22], a ring R is called a dual ring if every ideal of R is an annihilator. Let N be an R-module. A chain of R-submodules of length n is a sequence N i (0 i n) of R-submodules of N such that 0 = N 0 N 1 ::: N n = N . A composition series of N is a maximal chain, that is one in which no extra R-submodules can be inserted. Two composition series A 0 = 0 A 1 ::: A n = N and B 0 = 0 B 1 :::B t = N of an Rmodule N are said to be isomorphic (or equivalent) provided n = t and there exists a permutation of f1; 2; :::; ng such that A i =A i 1 = B (i) =B (i) 1 (isomorphic as an R-module) for all i = 1; :::; n. It is known that every two composition series for N are equivalent. The length of composition series of N is denoted by`(N ). A submodule K of an R-module M is called essential in M if, for every non-zero submodule L of M , we have K \ L 6 = 0. An R-module M is called uniform, if every non-zero submodule of M is essential in M . Let N; H be two submodules of R-module M . Then H is called a complement of N if H is maximal with respect to the property H \N = f0g. If N and H are complement of each other, then N and H are called mutual complement. An R-module N is called uniserial if its submodules are linearly ordered by inclusion. If R is uniserial as an R-module, then we call R is uniserial. Note that uniserial rings are in particular local rings. Commutative uniserial rings are also known as valuation rings. We call an R-module N serial if it is a direct sum of uniserial modules. The ring R is called serial if R is serial as an R-module.
For a graph G by E(G) and V (G) we denote the set of all edges and vertices, respectively. We recall that a graph is connected if there exists a path connecting any two distinct vertices. A graph G is said to be totally disconnected if it has no edges. The distance between two distinct vertices a and b, denoted by d(a; b), is the length of the shortest path connecting them (if such a path does not exist, then d(a; b) = 1). The diameter of a graph G, denoted by diam(G), is equal to supfd(a; b) : a; b 2 V (G)g. The eccentricity of a vertex a is de…ned as e(a) = maxfd(a; b) : b 2 V (G)g and the radius of G is given by rad(G) = minfe(x) : x 2 V (G)g. A vertex x of a connected graph G is a cut vertex of G if there are vertices a and b of G such that x is in every path from a to b (and x 6 = a, x 6 = b). Equivalently, for a connected graph G, x is a cut vertex of G if G fxg is not connected. The degree of a vertex x in a graph G is the number of edges incident with x. The degree of a vertex x is denoted by deg(x). Let r be a non-negative integer. The graph G is said to be r-regular, if the degree of each vertex is r. If a and b are two adjacent vertices of G, then we write a b. A vertex a of G is called end vertex, if deg(a) = 1. A graph is complete if it is connected with diameter less than or equal to one. We denote the complete graph on n vertices by K n . The girth of a graph G, denoted gr(G), is the length of a shortest cycle in G, provided G contains a cycle; otherwise; gr(G) = 1. A complete bipartite graph with part sizes m and n is denoted by K m;n . A star graph is a graph with a vertex adjacent to all other vertices and has no other edges. A clique of a graph is its maximal complete subgraph and the number of vertices in the largest clique of graph G, denoted by !(G), is called the clique number of G.

Basic properties of G(R)
In this section, we give some basic properties of the graph G(R) which are useful in the following sections. We begin with the following useful lemma.
Lemma 2. Let R be a ring and I; J two nontrivial ideals of R. Then the following statements hold.
(1) If I + J = R, then I and J are adjacent in G(R).
(2) If I J or J I, then I and J are adjacent in G(R).
The bene…t of studying the graph G(R) is that one may …nd some results about its known subgrapghs, for a ring R.

Remark 3.
(1) Let R be a ring. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial ideals of R and two distinct ideals I and J are adjacent if and only if I J or J I [1]. By Lemma 2, In(R) is a subgraph of G(R).
(2) Let R be a ring. The comaximal ideal graph, denoted by (R), is a graph whose vertices are proper ideals of R that are not contained in the Jacobson radical of R, and two vertices I and J are adjacent if and only if I + J = R [24]. By Lemma 2, (R) is a subgraph of G(R).
Proposition 4. Let R be a ring. Then the following statements hold.
(1) G(R) is a totally disconnected graph if and only if R has only one non-zero proper ideal.
(2) G(R) is a complete graph if and only if R is an Ikeda-Nakayama ring. Proof.
(1) One side is clear. To prove the other side, suppose that G(R) is a totally disconnected graph, with at least two vertices I; J. Since G(R) is totally disconnected, I \ J = f0g and I + J = R. Hence Lemma 1 gives I and J are adjacent, a contradiction. So R has only one non-zero proper ideal.
Lemma 5. Let R be a ring and max(R) = fM i : i 2 Kg. Then the following statements hold.
(2) Let I be a nonzero proper ideal of R and a …nite subset of K. We split the proof into three cases for I: Hence I and \ i2 M i are co-maximal, so I and \ i2 M i are adjacent by Lemma 2.
Case 3: Let = fi 2 : I M i g and = fi 2 : ann(I), hence ann(I) = ann(I) + i2 ann(M i ). Therefore Theorem 6. Let R be a ring. Then the following statements hold.
(1) If deg(I) is …nite for some ideal I of R, then R is an Artinian ring.
(3) By Lemma 5, every maximal ideal of R is adjacent to every other vertex of R. Hence (G(R)) = 1 and rad(G(R)) = 1.
Proof. Let R contain more than one non-zero proper ideal. Let I, J be two nonzero proper ideals of R. If I 2 max(R) or J 2 max(R), then I and J are adjacent by Lemma 2. Hence d(I; J) = 1. Suppose that I; J are not maximal. By Lemma 5, for each M 2 max(R), M is adjacent to I; J, hence d(I; J) 2. Hence G(R) is connected and diam(G(R)) 2 f0; 1; 2g.
Proof. If jmax(R)j 3, then gr(G(R)) = 3 by Lemma 5. Suppose jmax(R)j 2. We divide the proof in two cases: Hence M 1 ; M 2 are the only nonzero proper ideals of R, so gr(G(R)) = 1.
Case 2: Let (R; M ) be a local ring. If there exist non-zero proper (non-maximal) ideals I and J of R such that I & J, then gr(G(R)) = 3. Suppose, for each ideal I of R, there is no ideal J of R such that J I, hence each non maximal ideal of R is minimal, which gives gr(G(R)) = 1. Theorem 10. Let R be a ring. Then the following statements are equivalent: (1) G(R) contains an end vertex; is a local ring and each proper non-maximal ideal of R is minimal; ( Proof. (1) ) (2) Let I be an end vertex of G(R). If I is a maximal ideal of R, then jI(R)j = 2, because deg(I) = 1 and I is adjacent to every other vertex of G(R) by Lemma 5. Suppose that I is not maximal. By Lemma 5, I is adjacent to every maximal ideal of R. Hence R is a local ring. We show for each non-maximal ideal J 6 = 0 (J 6 = I) of R, J is minimal. Since I is only adjacent to the maximal ideal M of R, I \ J = f0g, I + J = M for each ideal J of R. Suppose that, there exists an ideal J of R such that J is not minimal. Hence there exists an ideal K of R such that K J. By the above argument, I K = M . By using modular law, J = K (I \ J). Hence J = K, a contradiction. So R is a local ring and each proper non-maximal ideal of R is minimal.
(2) ) (3) If R = M 1 M 2 , where I(R) = fM 1 ; M 2 g, then it is clear that G(R) is a star graph. If (R; M ) is a local ring and each proper non-maximal ideal of R is minimal, then M is adjacent to every other vertex of G(R) and two nonzero non-maximal ideals I; J of R are not adjacent (because I \ J = f0g and ann l (I) = ann l (J) = M ). Hence G(R) is a star graph.
(4) ) (5) By Corollary 9 and the proof of Theorem 8, G(R) is a star graph and so it is a bipartite graph. Theorem 11. Let R be a ring. Then the following statements are equivalent: (1) G(R) contains a cut vertex; (2) (i) (R; M ) is a local ring.
(ii) Each proper non-maximal ideal of R is minimal, maximal in M and`(R) = 3.
(iii) R has at least three non-trivial ideals.
Proof. Since G(R) 6 = K 2 , R has at least three non-trivial ideals.

3.
When is G(R) planar, complete or complete r-partite?
In this section, planar property, complete and complete r-partite property of G(R) are investigated. (1) G(R) is complete; (2) Soc(R) is simple; (3) R is a dual ring; (4) R is uniform; (5) R is an Ikeda-Nakayama ring.
(3) R = R 1 R 2 :::: R n (n 2 N), where (R i ; M i ) is local and Soc(R i ) is simple.
Proof. (1) ) (2) Let R be an Artinian ring. By [5,Theorem 8.7], R is isomorphic to the product of local Artinian rings R i with maximal ideals M i . We show that G(R i ) is complete. Let I; J be two nontrivial ideals of R i , then R 1 ::: R i 1 I R i+1 ::: R n and R 1 ::: R i 1 J R i+1 ::: R n are nontrivial ideals of R. As G(R) is complete, I and J are adjacent in R i . Therefore G(R i ) is complete.
(2) ) (1) Let I = I 1 ::: I n and J = J 1 ::: J n be two nontrivial ideals of R 1 ::: R n . Set S I = fi : I i is nontrivial g and S J = fi : J i is nontrivial g. If S I \ S J = ;, then I and J are adjacent. If S I \ S J 6 = ;, then by assumption, for each i 2 S I \ S J , I i and J i are adjacent in G(R i ). Hence I and J are adjacent.
Theorem 15. Let R be a ring with G(R) r-regular. Then either G(R) is complete or null graph.
Proof. Suppose G(R) is not null. By Theorem 6, R is an Artinian ring. Thus R = R 1 ::: R n by [5,Theorem 8.7], where (R i ; M i ) is a local Artinian ring. Toward a contradiction, assume that G(R) is not complete. Hence by Theorem 14, G(R i ) is not complete for some 1 i n. Thus Soc(R i ) is not minimal. Let I 1 and I 2 be two minimal ideals of R i . If H is a vertex of G(R i ) that is adjacent to I 1 , then I 1 H, by Lemma 12. Thus ann(H \ (I 1 I 2 )) = ann((H \ I 2 ) I 1 ) = M i = ann(I 1 I 2 ) ann(H). Therefore every vertex which is adjacent to I 1 is adjacent to I 1 I 2 too, in G(R i ). Moreover, I 1 and I 2 are not adjacent in G(R i ). This shows that deg(R 1 ::: R i 1 (I 1 I 2 ) R i+1 ::: R n ) > deg(R 1 ::: R i 1 I 1 R i+1 ::: R n ), a contradiction. Therefore G(R) is complete.
Theorem 16. Let R be an Artinian ring. Then R is a serial ring if and only if G( R I ) is complete for each ideal I of R. Proof. Let R be an Artinian ring. Then R = R 1 R 2 ::: R n , where (R i ; M i ) is a local Artinian ring. Assume that G( R I ) is complete for each ideal I of R. Let I i be an arbitrary ideal of R i . As R1 R2 ::: Rn R1 ::: Ri 1 Ii Ri+1 ::: Rn = Ri Ii , G( Ri Ii ) is complete by our assumption. Therefore Soc( Ri Ii ) is simple, by Lemma 13. This shows that R i is uniserial by [23, 55.1(1)]; and so R is serial.
Let R be a serial ring. Then R = R 1 R 2 ::: R n , where R i is a uniserial ring (and so local) for each 1 i n. As R i is Artinian for each 1 i n, Soc(R i =I i ) is simple for each proper ideal I i of R i . Therefore, G(R i =I i ) is complete by Lemma 13, for each 1 i n. Hence Theorem 14 implies that G( R I ) is complete for each ideal I of R.
In the following, we characterize the rings for which their Ikeda-Nakayama graph is planar. Proof. Let G(R) be a planar graph. Then by Lemma 17, R is Artinian. Hence R = R 1 R 2 , where (R i ; M i ) is a local ring for each i = 1; 2. If R i is not …eld for each i = 1; 2, then fR 1 0; R 1 M 2 ; M 1 R 2 ; 0 R 2 ; M 1 0g is a clique in G(R). Therefore, R 1 or R 2 is …eld. Let R 1 be …eld. We show that R 2 has only one non-zero proper ideal M 2 . Otherwise, assume that I is a non-zero ideal of R 2 such that I M 2 . Then fR 1 0; 0 R 2 ; R 1 M 2 ; R 1 I; 0 I; 0 M 2 g is a clique in G(R). This contradicts the planar property of G(R). Thus (R 2 ; M 2 ) is a local ring with the only non-zero proper ideal M 2 . Hence G(R) = K 4 and R is an Ikeda-Nakayama ring. If R 2 is a …eld (M 2 = 0), then G(R) = K 2 and R is an Ikeda-Nakayama ring.
(1) Let I be an ideal of R. By Lemma 17, R is Artinian. Hence Soc(R)\I 6 = f0g. Therefore ann(Soc(R)\I) = M . This implies that Soc(R) and I are adjacent.
(2) M 2 = 0, M = Rx + Ry, every proper ideal, di¤ erent from M , must be principal and of the form Rx, Ry, or R(x + ay), where a is an invertible element of R and G(R) is a star graph.
Proof. Let R be a ring with G(R) planar. By Lemma 21, the set of minimal generators for M has at most 2 elements. If M = Rx, then R is a principal ideal ring. This implies that I(R) = fM i : 1 i ng; where n is the smallest number such that M n = 0 and n 4. Thus (1) holds.
Hence, we have M = Rp + Rq and M 2 \ Rq = f0g; where p = x and q = x + y. Therefore, this case is similar to Case 2 and I(R) = fM; Rp; Rq; Rp 2 ; R(p + q); R(q + p 2 ); Rq Rp 2 = Soc(R)g: The converse is clear.
Theorem 26. Let R be a ring. If G(R) is a complete r-partite graph with part ; s V i (1 i r), then R is Artinian and one of the following statements hold: (1) G(R) is complete and R is an Ikeda-Nakayama ring.
(2) R is a local ring,`(Soc(R)) 2 and if I; J 2 V i , then I and J are cyclic local R-modules with common maximal submodule I \ J.
Proof. Since G(R) is complete r-partite, R is Artinian.
Suppose R is not local, so R = R 1 ::: R n , where R ; i s are local rings. Since G(R) is not complete, there exists i such that G(R i ) is not complete, by Theorem 14. Suppose, without lose of generality, G(R 1 ) is not complete, hence, there exist ideals I 1 and J 1 of R 1 such that they are not adjacent. So I 1 0 ::: 0 and J 1 0 ::: 0 are in the same part, say V 1 . Consider J 1 R 2 0 ::: 0 as an ideal of R. So I 1 0 ::: 0 is not adjacent to J 1 R 2 0::: 0 but J 1 0 ::: 0 is adjacent to J 1 R 2 0::: 0, which is a contradiction. Therefore G(R) is complete and (1) holds.
Suppose R is local. We show that`(Soc(R)) 2. Suppose, on the contrary, (Soc(R)) 3 and K 1 K 2 K 3 Soc(R), where K i is a minimal ideal of R, for each 1 i 3. It is clear that K 1 K 2 and K 3 are not adjacent. Let K 1 K 2 ; K 3 2 V j . Since K 2 is adjacent to K 1 K 2 , K 2 is adjacent to K 3 , which is a contradiction. Theorem 27. Let R be a ring. If !(G(R)) < 1, then the following statements hold.
(2) Since !(G(R)) is …nite, max(R) is …nite, by Lemma 5 (2). Let max(R) = fM 1 ; M 2 ; :::M t g and P (max(R)) be the power set of max(R). Let T X = \ T 2X T , for each X 2 P (max(R)). Then by Lemma 5, the subgraph of G(R) with vertex set fT X g f;; J(R)g is a complete subgraph of G(R), say G 0 (J(R) may be zero). So !(G(R)) 2 jmax(R)j 2. Now, let n = maxf`(M i ) : M i 2 max(R)g. Hence n =`(M j ) for some 1 j t and M j has the composition series 0 = N 0 N 1 ::: N n = M j ; for some submodules N s of M j (0 s n). Similar to the proof of Lemma 5, one can prove that for each 1 s n 1, every N s is adjacent to every other vertex of G 0 . Therefore V (G 0 ) [ fN s g n 1 s=1 is a clique in G(R), and thus !(G(R)) 2 jmax(R)j 2 + n 1 = 2 jmax(R)j + n 3.