Let $R$ be a commutative ring with nonzero  identity and $I$ be a proper ideal of $R$. The annihilator graph of $R$ with respect to $I$, which is denoted by  $AG_{I}(R)$, is the undirected graph with vertex-set $V(AG_{I}(R)) = \lbrace x\in R \setminus I :   xy \in I\ $ for  some$ \  y \notin I \rbrace$ and two distinct vertices $x$ and $y$ are adjacent if and only if    $A_{I}(xy)\neq A_{I}(x) \cup A_{I}(y)$, where $A_{I}(x) = \lbrace r\in R : rx\in I\rbrace$. In this paper, we study some basic properties of $AG_I(R)$, and we characterise  when  $ AG_{I}(R) $ is planar, outerplanar or a ring graph. Also, we study the graph $AG_{I}(\mathbb{Z}_{n}) $, where $Z_n$ is the ring of integers modulo $n$

Afkhami, Mojgan

Hoseini, Nesa

Khashyarmanesh, Kazem

English

Università del Salento: ESE - Salento University Publishing

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 36 (2016) no. 1, 1–10. doi:10.1285/i15900932v36n1p1The annihilator ideal graph of acommutative ringMojgan AfkhamiDepartment of Mathematics, University of Neyshabur,P.O.Box 91136-899, Neyshabur, Iranmojgan.afkhami@yahoo.comNesa HoseiniDepartment of Pure Mathematics, Ferdowsi University of Mashhad,P.O.Box 1159-91775, Mashhad, Irannesa.hoseini@gmail.comKazem KhashyarmaneshDepartment of Pure Mathematics, Ferdowsi University of Mashhad,P.O.Box 1159-91775, Mashhad, Irankhashyar@ipm.irReceived: 31.10.2014; accepted: 23.3.2015.Abstract. Let R be a commutative ring with nonzero identity and I be a proper idealof R. The annihilator graph of R with respect to I, which is denoted by AGI(R), is theundirected graph with vertex-set V (AGI(R)) = {x ∈ R \ I : xy ∈ I for some y /∈ I} andtwo distinct vertices x and y are adjacent if and only if AI(xy) 6= AI(x) ∪ AI(y), whereAI(x) = {r ∈ R : rx ∈ I}. In this paper, we study some basic properties of AGI(R), and wecharacterise when AGI(R) is planar, outerplanar or a ring graph. Also, we study the graphAGI(Zn), where Zn is the ring of integers modulo n.Keywords: Zero-divisor graph, Annihilator graph, Girth, Planar graph, Outerplanar, RinggraphMSC 2000 classification: primary 05C10, 05C99, secondary 13A99IntroductionLet R be a commutative ring with nonzero identity and let Z(R) be the set ofzero-divisors of R. Recently, there has been considerable attention in the litera-ture to associating graphs with algebraic structures (see [1, 9, 10, 11, 12]). Prob-ably the most attention has been to the zero-divisor graph, which is denoted byΓ(R). The set of vertices of Γ(R) is Z(R)∗ = Z(R)\{0}, and two distinct verticesx and y are adjacent if and only if xy = 0. The concept of the zero-divisor graphgoes back to Beck [5], who let all elements of R be vertices and was mainly inter-ested in colorings. The zero-divisor graph was introduced and studied by D. F.http://siba-ese.unisalento.it/ c© 2016 Universita` del Salento2 M. Afkhami, N. Hoseini and K. KhashyarmaneshAnderson and P. S. Livingston in [2]. For a recent survey article on zero-divisorgraphs see [3]. In [4], A. Badawi introduced the annihilator graph AG(R) for acommutative ring R. Let a ∈ R and annR(a) = {r ∈ R : ra = 0}. The annihila-tor graph of R is the (undirected) graph with vertices Z(R)∗, and two distinctvertices x and y are adjacent if and only if annR(xy) 6= annR(x) ∪ annR(y).Clearly, each edge of Γ(R) is an edge of AG(R), and so Γ(R) is a subgraph ofAG(R).In [13], S. P. Redmond introduced the zero-divisor graph of a commutativering R with respect to an ideal I of R. Let I be a proper ideal of R. Thezero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whosevertices are the set{x ∈ R \ I : xy ∈ I for some y /∈ I},and two distinct vertices x and y are adjacent if and only if xy ∈ I. In [13],among other results the relationship between the graphs ΓI(R) and Γ(R/I) wasexplored.In this article, we introduce the annihilator graph of R with respect to aproper ideal I of R, which is denoted by AGI(R). Let x ∈ R and AI(x) = {r ∈R : rx ∈ I}. The annihilator graph of R with respect to I is the (undirected)graph AGI(R) with vertices V (AGI(R)) = {x ∈ R \ I : xy ∈ I for some y /∈ I},and two distinct vertices x and y are adjacent if and only if AI(xy) 6= AI(x) ∪AI(y). In other words, two distinct vertices x and y are adjacent if and only ifthere exists r ∈ R \ I such that rxy ∈ I and rx, ry /∈ I. Note that ΓI(R) is asubgraph of AGI(R). Also if I = {0}, then we have AGI(R) = AG(R). We callthe graph AGI(R) the annihilator ideal graph.In Section 2 of this paper, we study some basic properties of AGI(R). Forinstance, we show that if AGI(R) is not identical to ΓI(R), then the girth of thegraph AGI(R) is at most 4 (see Theorem 2.7). In the third section, we studythe planarity of AGI(R). In Section 4, we obtain some results on the annihilatorideal graph of Zn, where Zn is the ring of integers modulo n for a positive integern. We also study situations under which the graphs AGI(Zn) and ΓI(Zn) areisomorphic.Now, we give a brief necessary background of graph theory. Let X be anundirected graph. We use the notation V (X) for the set of vertices of X. Also,for a vertex x ∈ V (X), N(x) denotes the set of vertices adjacent to x, and|N(X)| is called the degree of x. We say that the graph X is connected if thereis a path between each pair of distinct vertices of X. For two distinct verticesx and y, we define d(x, y) to be the length of a shortest path between x andy (d(x, y) = ∞ if there is no such path). The diameter of X is diam(X) =sup{d(x, y) : x and y are distinct vertices of X}. The girth of X is the lengthAnnihilator ideal graph 3of a shortest cycle in X, denoted by gr(X) ( gr(X) = ∞ if X has no cycles).A clique of a graph is any complete subgraph of the graph and the number ofvertices in a largest clique of X, denoted by ω(X), is called the clique numberof X. A planar graph is a graph that can be embedded in the plane, that is, itcan be drawn in the plane in such a way that its edges intersect only at theirendpoints. Kuratowski provided a characterization of planar graphs, which isnow known as Kuratowski’s Theorem: A finite graph is planar if and only if itdoes not contain a subdivision of K5 or K3,3 [6, Theorem 9.10]. Suppose thatX is a graph with n vertices and q edges. Also, assume that C is a cycle ofX. A chord in X is any edge of X joining two non-adjacent vertices in C. Aprimitive cycle is a cycle without chords. Moreover, we say that a graph Xhas the primitive cycle property (PCP) if any two primitive cycles intersect inat most one edge. The free rank of X, denoted by frank(X), is the number ofprimitive cycles of X. Also, the number rank(X) = q − n + r, where r is thenumber of connected components of X, is called the cycle rank of X. The cyclerank of X can be expressed as the dimension of the cycle space of X. These twonumbers satisfy the inequality rank(X) ≤ frank(X), as is seen in [7, Proposition2.2]. The precise definition of a ring graph can be found in Section 2 of [7]; theauthors showed that, for the graph X, the following conditions are equivalent:(i) X is a ring graph,(ii) rank(X) = frank(X),(iii) X satisfies PCP and X does not contain a subdivision of K4 as a subgraph.Clearly ring graphs are planar. An undirected graph is an outerplanar graphif it can be drawn in the plane without crossings in such a way that all of thevertices belong to the unbounded face of the drawing. There is a characterizationof outerplanar graphs that says a graph is outerplanar if and only if it does notcontain a subdivision of the complete graph K4 or the complete bipartite graphK2,3 [8, Theorem 11.10]. Clearly, every outerplanar graph is planar. Let G andH be graphs. We use the notations G = H and G ∼= H to denote identical andisomorphic graphs, respectively.As usual, Z and Zn will denote the rings of integers and integers modulo n,respectively.1 Basic properties of AGI(R)In this section, we investigate the basic properties of AGI(R). Also, westudy the relations between the graphs AGI(R) and AG(R/I). The followingproposition immediately follows from the definition of the graph AGI(R).4 M. Afkhami, N. Hoseini and K. KhashyarmaneshProposition 1.1. Let I be a proper ideal of R. Then AGI(R) = ∅ if andonly if I is a prime ideal.Proposition 1.2. Let x + I and y + I be distinct elements in R/I. Thenx+I is adjacent to y+I in AG(R/I) if and only if x is adjacent to y in AGI(R).Proof. Since AI(xy) 6= AI(x)∪AI(y) if and only if annR/I(xy+I) 6= annR/I(x+I) ∪ annR/I(y + I), we have x+ I is adjacent to y + I in AG(R/I) if and onlyif x is adjacent to y in AGI(R). QEDTheorem 1.3. If x+ i is adjacent to y+ i in AGI(R), for some i ∈ I, thenall elements of x+ I and y + Iare adjacent in AGI(R).Proof. Assume that x+ i is adjacent to y+ i, for some i ∈ I. Hence there existsr ∈ R \ I such that r(x+ i)(y + i) ∈ I, r(x+ i) /∈ I, and r(y + i) /∈ I. Then, forevery j, k ∈ I, we have r(x + j)(y + k) ∈ I, r(x + j) /∈ I, and r(y + k) /∈ I. Soall vertices of x+ I and y + I are adjacent in AGI(R). QEDIn the following example, we show that if AG(R/I) ∼= AG(S/J), where Iand J are ideals of the rings R and S, respectively, then the graphs AGI(R)and AGJ(S) are not necessarily isomorphic.Example 1.4. Let R = Z6 × Z3 and I = 0 × Z3. Then AG(R/I) ∼= AG(Z6)with vertex-set {(2, 0), (3, 0), (4, 0)}, where (3, 0) is adjacent to both vertices(2, 0) and (4, 0). Let S = Z24 and J =< 8 >. Then AG(S/J) ∼= AG(Z8) withvertex-set {2, 4, 6}, where the vertex 4 is adjacent to the vertices 2 and 6. Nowthe graph AGI(R) is pictured in Figure 1 while AGJ(S) is a complete graph.Hence AG(R/I) ∼= AG(S/J), but the graphs AGI(R) and AGJ(S) are notisomorphic.(2,0)(3,0)(4,0)(2,1)(3,1)(4,1)(2,2)(3,2)(4,2)Figure 1Annihilator ideal graph 5Let {aλ}λ∈Λ ⊆ R be a set of coset representatives of the vertices of AG(R/I).In [13], S. P. Redmond defined the concept of a column and a connected column.Here, we call the subset aλ+I = {aλ+i : i ∈ I} a column of AGI(R). Moreover,if there exists r ∈ R \ I such that raλ /∈ I and ra2λ ∈ I, then we say that aλ + Iis a connected column of AGI(R).Now the following two theorems follow immediately from the previous para-graph.Theorem 1.5. If a + I is a connected column of AGI(R), then a + I is acomplete subgraph of AGI(R) and ω(AGI(R)) ≥ |I|.Theorem 1.6. If AGI(R) has a connected column and |V (AG(R/I))| ≥ 2,then ω(AGI(R)) ≥ |I|+ 1.In the next theorem, we study the girth of the graph AGI(R).Theorem 1.7. Let AGI(R) 6= ΓI(R). Then gr(AGI(R)) ≤ 4.Proof. Since AGI(R) 6= ΓI(R), there exist adjacent vertices x and y such thatxy /∈ I. Thus there exists 1 6= r ∈ R \ I such that rxy ∈ I, rx /∈ I and ry /∈ I.Now, if r /∈ {x, y}, then it is easy to see that r is a vertex of AGI(R) whichis adjacent to both vertices x and y, and so we have the cycle x − r − y − x.Without loss of generality, we may assume that r = x. Then clearly x 6= x2,y 6= xy and we have the cycle x− y − x2 − xy − x. So the result holds. QED2 Planar, outerplanar and ring graph annihilator idealgraphsWe begin this section with the following theorem.Theorem 2.1. Let |V (AG(R/I))| = 1. Then AGI(R) is planar if and onlyif |I| 6 4.Proof. Suppose that V (AG(R/I)) = {x + I}. Clearly x2 ∈ I and x + I is aconnected column of AGI(R). Hence, by Theorem 1.5, AGI(R) is isomorphicto the complete graph K|I|. Therefore AGI(R) is planar if and only if |I| ≤4. QEDLemma 2.2. Let |V (AG(R/I))| > 2 and |I| ≥ 3. Then AGI(R) is notplanar.Proof. Suppose that a+I is a vertex of AG(R/I). Note that AG(R) is connectedsince Γ(R) is connected [2, Theorem 2.3], and Γ(R) is a subgraph of AG(R)with V (Γ(R)) = V (AG(R)). Now since |V (AG(R/I))| > 2 and AG(R/I) isconnected, there exists b ∈ R \ I such that the vertices a and b are adjacent6 M. Afkhami, N. Hoseini and K. Khashyarmaneshin AGI(R). Hence AGI(R) has a subgraph isomorphic to K3,3 with vertex-set{a, a + x1, a + x2} ∪ {b, b + x1, b + x2}, where x1 and x2 are distinct elementsin I \ {0}. So AGI(R) contains a subgraph which is isomorphic to K3,3. ThusAGI(R) is not planar. QEDTheorem 2.3. Let |V (AG(R/I))| = 3 and |I| = 2. Then AGI(R) is notplanar if and only if AG(R/I) is complete such that at least one of its verticesis a connected column in AGI(R).Proof. First assume that AG(R/I) is a complete graph with vertices a+I, b+Iand c+ I. Without loss of generality, we may assume that a+ I is a connectedcolumn. Then AGI(R) has a subgraph isomorphism to K3,3 with vertex-set{b, b+x, a}∪{c, c+x, a+x}, where x ∈ I \{0}. Hence AGI(R) contains a copyof K3,3, which means that it is not planar.For the converse statement, assume to the contrary that if AG(R/I) is com-plete, then none of its vertices is a connected column, or AG(R/I) is not com-plete. If AG(R/I) is complete and no vertex is a connected column, then AGI(R)is pictured in Figure 2, which is planar, and this is a contradiction.b + xa + xcc + x baFigure 2Now assume that AG(R/I) is not a triangle, say that a + I and c + I arenot adjacent. Then AGI(R) is pictured in Figure 3, which is planar, and this isa contradiction.Hence the result holds. QEDTheorem 2.4. Let |V (AG(R/I))| > 4 and |I| = 2. Then AGI(R) is notplanar if and only if there exists a subdivision of the complete graph K3 inAG(R/I).Proof. First assume that there exists a subdivision of the complete graph K3in AG(R/I). Suppose that a+ I, b+ I, c+ I and d+ I are distinct vertices inAG(R/I). Now if these vertices form a square in AG(R/I), then AGI(R) has aAnnihilator ideal graph 7a + xb + xc c + xbaFigure 3subgraph isomorphic to K3,3 with vertex-set {a, a + x, c} ∪ {b, b + x, d}, wherex ∈ I \{0}. Hence AGI(R) contains a subgraph isomorphic to K3,3. So AGI(R)is not planar. If a, b and c form a triangle in AGI(R/I), then, in view of Figure4, the graph AGI(R) contains a subdivision of K3,3. So AGI(R) is not planar.aa +xc + xbcdb + xFigure 4The converse statement is clear. QEDTheorem 2.5. Let |V (AG(R/I))| = 1. Then |I| ≤ 3 if and only if AGI(R)is a ring graph and an outerplanar graph.Proof. Since |V (A(R/I))| = 1, then, in view of the proof of Theorem 2.1,AGI(R) is isomorphic to K|I|. If |I| > 4, then AGI(R) contains a subgraphwhich is isomorphic to K4. So AGI(R) is neither a ring graph nor an outerpla-nar graph.The converse statement is clear. QEDSince the graph AGI(R) is isomorphic to AG(R) whenever |I| = 1, in thefollowing theorem, we verify the case that |I| > 2 and |V (A(R/I))| ≥ 2.Theorem 2.6. Assume that |I|, |V (AG(R/I))| ≥ 2, and at least one of thesets V (AG(R/I)) or I has three elements. Then AGI(R) is neither outerplanarnor a ring graph.8 M. Afkhami, N. Hoseini and K. KhashyarmaneshProof. First, suppose that |I| ≥ 3. Then the set of vertices {a, a+ x1, a+ x2} ∪{b, b+x1}, where x1 and x2 are distinct nonzero elements of I and a, b are distinctelements of R \ I, forms the graph K2,3. So AGI(R) contains a subgraph whichis isomorphic to K2,3. Assume that |V (AG(R/I))| ≥ 3 and that a+I, b+I, c+Iare distinct vertices of AG(R/I). Since AG(R/I) is connected, we may assumethat a+ I and c+ I are adjacent to b+ I. Thus the vertices a and c are adjacentto b in AGI(R). Therefore the set of vertices {a, a+x1, c}∪{b, b+x1} forms thegraph K2,3. Hence AGI(R) contains a subgraph which is isomorphic to K2,3. SoAGI(R) is neither outerplanar nor a ring graph. QEDIn the following theorem, we study the outerplanar and ring graph annihi-lator ideal graphs in the remaining case |I| = 2 and |V (A(R/I))| = 2.Theorem 2.7. Let |I| = |V (AG(R/I))| = 2. Then AGI(R) is neither out-erplanar nor a ring graph if and only if there exist two connected columns inAGI(R).Proof. Note that AGI(R) is connected since ΓI(R) is connected [13, Theorem2.4], and ΓI(R) is a subgraph of AGI(R) with V (ΓI(R)) = V (AGI(R)). We have|I| = |V (AG(R/I))| = 2, and |V (AGI(R))| = 4. So it is easy to see that thereexist two connected columns in AGI(R) if and only if AGI(R) is isomorphic toK4. Therefore the result holds. QED3 Annihilator ideal graph of ZnIn this section, we assume that n is a positive integer and p, q are distinctprime numbers that divide n. Also, for x ∈ Zn, < x > denotes the ideal gener-ated by x in Zn.Theorem 3.1. Let R = Zn and I =< pq >. Then AGI(Zn) = ΓI(Zn).Proof. Let x and y be adjacent vertices of AGI(Zn). Then there exists r ∈ R\ Isuch that rxy ∈ I, rx ∈ R \ I, and ry ∈ R \ I. It is enough to show that xy ∈ I.Since rxy ∈ I and p | pq, we have p | rxy, which implies that p | r, p | x or p | y,and q | r, q | x or q | y. Let p | r. Since pq - rx and pq - ry, we have q | r. Thuspq | r2, and this implies that pq | r. So r ∈ I, which is a contradiction. Let p | x.Then q | y. So xy ∈ I. The case that p | y is obtained in a similar way. QEDTheorem 3.2. Let R = Zn and I =< p2 >. Then AGI(Zn) = ΓI(Zn).Proof. The proof is similar to the proof of Theorem 3.1. QEDTheorem 3.3. Let R = Zpn and I =< pn−i >, for some 1 ≤ i ≤ n − 1.Then AGI(R) is a complete graph.Annihilator ideal graph 9Proof. Let x and y be distinct vertices of V (AGI(Zpn)). Then xx′ ∈ I andyy′ ∈ I, for some vertices x′, y′ ∈ V (AI(Zpn)). Hence pn−i | xx′ and pn−i | yy′.Set α = Max{α′ ∈ N : pα′ | x} and β = Max{β′ ∈ N : pβ′ | y}. Then we havepα+β | xy. Now, if α+ β > n− i, then xy ∈ I, which implies that x is adjacentto y. Let α + β < n − i and put r = pn−i−(α+β). This implies that rxy ∈ I,and we have that rx ∈ R \ I. Assume that rx ∈ I. Then pn−i | pn−i−(α+β)pαa,where there exists a ∈ Z such that pαa = x. Thus pβ | a, and hence pα+β | x.Therefore we have α + β ≤ α which implies that β = 0 and pn−i | y′. This is acontradiction. QEDTheorem 3.4. Suppose that R = Zn and I = 〈k〉, where k ∈ Zn. If thereexist vertices a, b, k ∈ Zn such that gcd(a, k) = gcd(b, k), then deg(a) = deg(b)and N(a) \ {b} = N(b) \ {a}.Proof. Let s 6= b be an adjacent vertex to a. Then there exists r ∈ R \ I suchthat rsa ∈ I, rs /∈ I, and ra /∈ I. Thus k | rsa, k - rs, and k - ra. In orderto show that s ∈ N(b), it is enough to prove that k | rsb, and k - rb. Setd = gcd(a, k) = gcd(b, k). Then we havekd| rsad, and sokd| rs, k | rsd andk | rsb. Now, assume on the contrary that k | rb. Then kd| r bd, and sokd| r,kd| rad, and k | ra, which is contradiction. Therefore s ∈ N(b) \ {a}, and so theresults hold. QEDTheorem 3.5. Suppose that R = Zn and I = 〈k〉, where k ∈ Zn. If thereexists an integer d > 1 such that d2 | k, then gr(AGI(Zn)) = 3.Proof. Consider two distinct integers s1, s2 such that gcd(s1, k) = gcd(s2, k) =1. Then gcd(s1d, k) = gcd(s2d, k) = d and, by Theorem 3.4, N(s1d) \ {s2d} =N(s2d) \ {s1d}. Now it is easy to see that s1d is adjacent to s2d and there existat least three vertices in AGI(Zn), which completes the proof. QEDAcknowledgements. The authors are deeply grateful to the referee forcareful reading of the manuscript and helpful suggestions.References[1] S. Akbari, H. R. Maimani, S. Yassemi, When a zero-divisor graph is planar or acomplete r-partite graph, J. Algebra 270 (2003), 169-180.[2] D. F. Anderson, P. S. Livingston, The zero-divisor graph of commutative ring, J.Algebra 217(1999), 434-447.10 M. Afkhami, N. Hoseini and K. Khashyarmanesh[3] D. F. Anderson, M. C. Axtell, J. A. Stickles JR, Zero-divisor graphs in commu-tative ring, In:M. Fontana, S. E. Kabbaj, B. Olberding, I. Swanson, eds. CommutativeAlgebra, noetherian and non-noetherian perspectives. New York: Springer-Verlag, pp.23-45 (2011).[4] A. Badawi,The annihilator ideal graph of a commutative ring, Comm. Algebra 42 (2014),108-121.[5] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226.[6] J. A. Bondy, U. S. R. Murty, Graph theory with applications, American Elsevier, NewYork, 1976.[7] I. Gitler, E. Reyes, R. H. Villarreal, Ring graphs and complete intersection toricideals, Discrete Math. 310 (2010), 430-441.[8] F. Harary, Graph theory, Addision-Wesley, Reading, MA, 1972.[9] C. Hung-Jen, Classification of rings with projective zero-divisor graphs, J. Algebra 319(2008), 2789-2802.[10] W. Hsin-ju, Zero-divisor graphs of genus one, J. Algebra 304 (2006), 666-678.[11] C. Wickham, Rings whose zero-divisor graphs have positive genus, J. Algebra 321 (2009)377-383.[12] H. R. Maimani, M. R. pournaki, A. Tehranian, S. Yassemi, Graphs attached to ringsrevisited, Arab. J. Sci. Eng. 36 (2011), 997-1012.[13] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra31 (2003), 4425-4443.

The annihilator ideal graph of a commutative ring

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The annihilator ideal graph of a commutative ring

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