On the crosscap of the annihilating-ideal graph of a commutative ring

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1 Palestine Journal of Mathematics Vol. 7()(08), 5 60 Palestine Polytechnic University-PPU 08 On the crosscap of the annihilating-ideal graph of a commutative ring K. Selvakumar and P. Subbulakshmi Communicated by Ayman Badawi MSC 00 Classifications: Primary 05C5; 05C75; Secondary 3A5, 3M05. Keywords and phrases: finite ring, planar, crosscap, local ring, annihilating-ideal graph. The authors would like to thank the referee for careful reading of the manuscript and helpful comments. The work is supported by the UGC Major Research Project (F. No. 4-8/03(SR)) awarded to the first author by the University Grants Commission, Government of India. Abstract Let R be a non-domain commutative ring with identity and A (R) be the set of non-zero ideals with non-zero annihilators. We call an ideal I of R, an annihilating-ideal if there exists a non-zero ideal I of R such that I I = (0). The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A (R) and two distinct vertices I and I are adjacent if and only if I I = (0). In this paper, we characterize all commutative Artinian non-local rings R for which AG(R) is planar and the crosscap of AG(R) is one. Introduction The study of algebraic structures, using the properties of graphs, became an exciting research topic in the past twenty years, leading to many fascinating results and questions. In the literature, there are many papers assigning graphs to rings, groups and semigroups, see [3, 4, 5,, 0, 3, 3, 4, 5]. In ring theory, the structure of a ring R is closely tied to ideal s behavior more than elements and so it is deserving to define a graph with vertex set as ideals instead of elements. Recently M. Behboodi and Z. Rakeei [3, 4] have introduced and investigated the annihilatingideal graph of a commutative ring. For a non-domain commutative ring R, let A (R) be the set of non-zero ideals with non-zero annihilators. We call an ideal I of R, an annihilating-ideal if there exists a non-zero ideal I of R such that I I = (0). The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A (R) and two distinct vertices I and I are adjacent if and only if I I = (0). Several properties of AG(R) were studied by the authors in [,, 3, 4, 6, 7]. By a graph G = (V,E), we mean an undirected simple graph with vertex set V and edge set E. A graph in which each pair of distinct vertices is joined by the edge is called a complete graph. We usek n to denote the complete graph withnvertices. Anr-partite graph is one whose vertex set can be partitioned into r subsets so that no edge has both ends in any one subset. A complete r-partite graph is one in which each vertex is joined to every vertex that is not in the same subset. The complete bipartite graph (-partite graph) with part sizes m and n is denoted by K m,n. The girth of G is the length of a shortest cycle in G and is denoted by gr(g). If G has no cycles, we define the girth of G to be infinite. The corona of two graphs G and G is the graph G G formed from one copy ofg and V(G ) copies ofg, where thei th vertex ofg is adjacent to every vertex in thei th copy ofg. A graph G is said to be planar if it can be drawn in the plane so that its edges intersect only at their ends. A subdivision of a graph is a graph obtained from it by replacing edges with pairwise internally-disjoint paths. A remarkably simple characterization of planar graphs was given by Kuratowski in 930. Kuratowski s Theorem says that a graphgis planar if and only if it contains no subdivision ofk 5 ork 3,3 (see [6, p.53]). A minor of G is a graph obtained from G by contracting edges in G or deleting edges and isolated vertices in G. A classical theorem due to K. Wagner [30] states that a graph G is planar if and only if G does not have K 5 or K 3,3 as a minor. It is well known that if G is a minor of G, then γ(g ) γ(g).

2 5 K. Selvakumar and P. Subbulakshmi The main objective of topological graph theory is to embed a graph into a surface. By a surface, we mean a connected two-dimensional real manifold, i.e., a connected topological space such that each point has a neighborhood homeomorphic to an open disk. It is well known that any compact surface is either homeomorphic to a sphere, or to a connected sum of g tori, or to a connected sum of k projective planes (see [, Theorem 5.]). We denote S g for the surface formed by a connected sum of g tori, and N k for the one formed by a connected sum of k projective planes. The numberg is called the genus of the surfaces g and k is called the crosscap of N k. When considering the orientability, the surfaces S g and sphere are among the orientable class and the surfaces N k are among the non-orientable one. In this paper, we mainly focus on the non-orientable cases. A simple graph which can be embedded in S g but not in S g is called a graph of genus g. Similarly, if it can be embedded in N k but not in N k, then we call it a graph of crosscap k. The notations γ(g) and γ(g) are denoted for the genus and crosscap of a graph G, respectively. It is easy to see that γ(h) γ(g) and γ(h) γ(g) for all subgraph H of G. For details on the notion of embedding of graphs in surface, one can refer to A. T. White [3]. Planarity ofag(r) The main goal of this section is to determine all commutative Artinian non-local rings R for which AG(R) is planar. Theorem.. [6] Let R = F F F n be a commutative ring with identity where each F i is a field and n. Then AG(R) is planar if and only if n = or n = 3. Theorem.. Let R = R R R n be a commutative ring with identity where each (R i,m i ) is a local ring with m i {0} and n. Let n i be the nilpotency of m i. Then AG(R) is planar if and only if n = and one of the following condition holds: (i) n =,n = 3 andm is the only non-trivial ideal inr andm,m are the only non-trivial ideals in R ; (ii) n = 3, n = and m,m are the only non-trivial ideals in R and m is the only nontrivial ideal in R ; (iii) n = n = and m and m are the only non-trivial ideal in R and R respectively. Proof. Suppose that n =. Then R = R R and hence the proof follows from Fig... m R m (0) (0) R m m R (0) (0) m m m m (0) (0) m m R R m (0) m R m R m m m (0) R R (0) (a) n = and n = 3 (b)n = and n = Fig.: AG(R R ) Conversely, assume that AG(R) is planar. Suppose that n >. Consider the non-trivial ideals x = m n (0) (0) (0), x = (0) m n (0) (0), x 3 = (0) (0) m n 3 3 (0), y = m m (0) (0), y = (0) m m 3 (0), y 3 = m (0) m 3 (0) in R. Then x i y j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence n =.

3 Crosscap of the annihilating-ideal graph of a commutative ring 53 Suppose that n > and n >. Consider the non-trivial ideals u = m n (0), u = (0) m n, u 3 = m n m n, u 4 = m (0), u 5 = (0) m in R. Then u i u j = (0) for every i j and so K 5 is a subgraph of AG(R), a contradiction. Hence n = or n =. Without loss of generality, we assume that n =. Suppose that n > 3. Consider the non-trivial ideals a = (0) m n, a = m m n, a 3 = m (0),b = (0) m,b = m m,b 3 = (0) m n in R. Then a i b j = (0) for every i,j and so K 3,3 is a subgraph ofag(r), a contradiction. Hence n 3. Suppose that n = 3. Let I be any non-trivial ideal in R with I m,m. Consider the non-trivial ideals x = (0) m, x = m m, x 3 = m (0), y = (0) m, y = m m, y 3 = (0) I in R. Then x i y j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence m,m are the only non-trivial ideals in R. Let I be any non-trivial ideal in R with I m. Consider the non-trivial ideals u = m m,u = (0) m,u 3 = m (0),u 4 = I (0),u 5 = (0) m inr. Thenu i u j = (0) for every i j and sok 5 is a subgraph ofag(r), a contradiction. Hence m is the only non-trivial ideal in R. Suppose that n =. Let I be any non-trivial ideal in R with I m. Consider the non-trivial ideals x = (0) m,x = (0) I,x 3 = m m, x 4 = m I, x 5 = m (0) in R. Then x i x j = (0) for every i j and so K 5 is a subgraph of AG(R), a contradiction. Hence m is the only non-trivial ideal in R. Similarly one can prove that m is the only non-trivial ideal in R. Lemma.3. Let (R,m) be a local ring. If dim(m/m ) = and for some positive integer t, m t = (0), then the set of all non-trivial ideals of R is the set {m i : i < t}. Proof. Since dim(m/m ) =, by Nakayama s lemma, m = Rx for some x R. Now, let I be a non-trivial ideal of R. Since m t = (0), there exists a natural number i t such that I m i and I m i+. Let a I\m i+. We have a = bx i for some b R. If b m, then a m i+, a contradiction. Thus b is an unit. Hence x i I. This implies that I = x i = m i, as desired. Thus, the set of all non-trivial ideals of R is the set {m i : i < t}. The next Proposition has a crucial role in this paper. Proposition.4. If (R,m) is a local ring and there is an ideal I of R such that I m i for every i, then R has at least three distinct non-trivial ideals J, K and L such that J, K, L m i for every i. Proof. Assume that R has an ideal I such that I m i for every i. Then by Lemma.3, dim(m/m ) = n. Therefore, by Nakayama s Lemma, we can find x,x,...,x n / m such that m = x,x,...,x n. Thus, Rx, Rx and R(x + x ) are the distinct non-trivial ideals with desired properties. Theorem.5. Let R = R R R n F F F m be a commutative ring with identity where each (R i,m i ) is a local ring with m i {0} and F j is a field, n,m. Let n i be the nilpotency of m i. Then AG(R) is planar if and only if one of the following condition holds: (i) R = R F F, n = and m is the only non-trivial ideal in R ; (ii) R = R F and one of the following holds: (a) n = and m is the only non-trivial ideal in R ; (b) n = 3 and m,m are the only non-trivial ideals in R ; (c) n = 4 and m,m,m3 are the only non-trivial ideals in R. Proof. Suppose R = R F F, n = and m is the only non-trivial ideal in R. Then AG(R) is isomorphic to the graph given in Fig.(a). Hence AG(R) is planar. Suppose R = R F, n = and m is the only non-trivial ideal in R. Then AG(R) is isomorphic to the graph given in Fig.(d). Hence AG(R) is planar. Supposen = 3 andm,m are the only non-trivial ideals inr. ThenAG(R) is isomorphic to the graph given in Fig.(c). Hence AG(R) is planar. Supposen = 4 andm,m,m3 are the only non-trivial ideals inr. ThenAG(R) is isomorphic to the graph given in Fig.(b). Hence AG(R) is planar.

4 54 K. Selvakumar and P. Subbulakshmi m F (0) m (0) (0) m (0) F R (0) F m F F (0) F (0) R F (0) (0) (0) F m (0) m 3 F m 3 (0) R (0) m F m F (0) F F R (0) (0) (0) F m (0) (a)ag(r F F ) (b)ag(r F ) with n = 4 m m (0) m (0) F (0) F R (0) m F (0) F m (0) m F R (0) (c)ag(r F ) with n = 3 (d)ag(r F ) with n = Fig. Conversely, assume that AG(R) is planar. Suppose n >. Consider the non-trivial ideals x = m n (0) (0) (0),x = (0) m n (0) (0),x 3 = m n m n (0) (0),y = (0) (0) F (0) (0),y = m (0) F (0) (0), y 3 = (0) m F (0) (0) in R. Then x i y j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence n =. Suppose that m >. Consider the non-trivial ideals u = m (0) (0) (0), u = m F (0) (0) (0),u 3 = (0) F (0) (0) (0),v = (0) (0) F (0) (0),v = (0) (0) (0) F 3 (0),v 3 = (0) (0) F F 3 (0) in R. Thenu i v j = (0) for everyi,j and sok 3,3 is a subgraph ofag(r), a contradiction. Hence m. Suppose that m =. Let n >. Consider the non-trivial ideals a = R (0) (0), a = m (0) (0), a 3 = m (0) (0), b = (0) F (0), b = (0) (0) F, b 3 = (0) F F in R. Then a i b j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence n =. Suppose that I is any non-trivial ideal in R with I m. Consider the non-trivial ideals d = R (0) (0), d = m (0) (0), d 3 = I (0) (0), e = (0) F (0), e = (0) (0) F, e 3 = (0) F F in R. Then d i e j = (0) for every i,j and so K 3,3 is a subgraph ofag(r), a contradiction. Hence m is the only non-trivial ideal in R. Suppose that m =. Let n > 4. Consider the non-trivial ideals x = m n (0), x = m n (0), x 3 = m n 3 (0), y = (0) F, y = m n F, y 3 = m n F in R. Then x i y j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence n 4. Assume that n =. Suppose there is an ideal I of R such that I m i for i =,. Then by Proposition.4, R has at least three distinct non-trivial ideals I, I and I 3 such that I, I, I 3 m. Consider the non-trivial ideals d = m (0), d = I (0), d 3 = I (0), e = (0) F, e = m F, e 3 = I F in R. Then d i e j = (0) for every i,j and so K 3,3 is a subgraph ofag(r), a contradiction. Hence m is the only non-trivial ideal in R.

5 Crosscap of the annihilating-ideal graph of a commutative ring 55 Assume that n = 3. Suppose there is an ideal I of R such that I m i for all i =,, 3. Then by Proposition.4, R has at least three distinct non-trivial ideals I, I and I 3 such that I, I, I 3 m i for all i =,. Consider the non-trivial ideals u = m (0), u = I (0), u 3 = I (0), v = (0) F, v = m F, v 3 = m (0) in R. Then u iv j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence m,m are the only non-trivial ideals in R. Assume that n = 4. Let I be any non-trivial ideal in R with I m,m,m3. Consider the non-trivial ideals q = m (0), q = m (0), q 3 = I (0), w = (0) F, w = m 3 F, w 3 = m 3 (0) in R. Then q iw j = (0) for every i,j and so K 3,3 is a subgraph of AG(R), a contradiction. Hence m,m,m3 are the only non-trivial ideals in R. 3 Crosscap ofag(r) The main goal of this section is to determine all commutative Artinian non-local rings R for which AG(R) has crosscap one. The following two results about the crosscap formulae of a complete graph and a complete bipartite graph are very useful in the subsequent sections. Lemma 3.. Let m, n be integers and for a real number x, x is the least integer that is greater than or equal to x. Then { 6 (n 3)(n 4) if n 3 and n 7 (i) γ(k n ) = 3 if n = 7 In particular, γ(k n ) = if n = 5, 6. (ii) γ(k m,n ) = (m )(n ), where n,m >. In particular, γ(k 3,n ) = if n = 3, 4. Theorem 3.. Let R = F F F n be a commutative ring with identity where each F i is a field and n >. Then γ(ag(r)) = if and only if n = 4. Proof. Assume that γ(ag(r)) =. Suppose that n > 4. Consider the non-trivial ideals x = F (0) (0) (0) (0) (0), x = (0) F (0) (0) (0) (0), x 3 = F F (0) (0) (0) (0), y = (0) (0) F 3 (0) (0) (0), y = (0) (0) (0) F 4 (0) (0), y 3 = (0) (0) (0) (0) F 5 (0), y 4 = (0) (0) F 3 F 4 (0) (0),y 5 = (0) (0) F 3 (0) F 5 (0) inr. Then x i y j = (0) for every i,j and so K 3,5 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >, a contradiction. Hence by Theorem., n = 4. a b b 3 a x x 4 a 3 b a x 3 x b a Fig 3.: Projective embedding of AG(F F F 3 F 4 ) Conversely, suppose thatn = 4. Consider all the non-trivial idealsa = F (0) (0) (0),a = (0) F (0) (0),a 3 = F F (0) (0),b = (0) (0) F 3 (0),b = (0) (0) (0) F 4, b 3 = (0) (0) F 3 F 4,x = F (0) (0) F 4,x = (0) F (0) F 4,x 3 = F (0) F 3 (0),

6 56 K. Selvakumar and P. Subbulakshmi x 4 = (0) F F 3 (0),x 5 = (0) F F 3 F 4,x 6 = F F (0) F 4,x 7 = F F F 3 (0), x 8 = F (0) F 3 F 4 inr. Thena i b j = (0) for everyi,j and sok 3,3 is a subgraph ofag(r). Therefore by Lemma 3.,γ(AG(R)). The embedding given in Fig 3. explicitly shows that γ(ag(r))=. Theorem 3.3. Let R = R R R n be a commutative ring with identity, where each (R i,m i ) is a local ring with m i {0} and n >. Let n i be the nilpotency of m i. If AG(R) is non-planar, then γ(ag(r)) >. Proof. Assume that AG(R) is non-planar. Suppose that n >. Consider the non-trivial ideals x = m n (0) (0) (0),x = (0) m n (0) (0),x 3 = m n m n (0), y = (0) (0) m 3 (0) (0), y = m (0) m 3 (0) (0), y 3 = (0) m m 3 (0) (0),y 4 = m m m 3 (0) (0),y 5 = (0) (0) R 3 (0) (0) in R. Then x i y j = (0) for every i,j and so K 3,5 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >. Hence n =. Suppose that n > and n >. Consider the non-trivial ideals a = m n (0), a = (0) m n,a 3 = m n m n,b = m (0),b = (0) m,b 3 = m m,b 4 = m n m, b 5 = m m n in R. Then a i b j = (0) for all i,j and so K 3,5 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >. Hence n = or n =. Without loss of generality, we assume that n =. Suppose that n > 3. Consider the set Ω = {e,...,e 9 }, where e = m m n, e = m m n,e 3 = m m,e 4 = m (0),e 5 = (0) m n,e 6 = (0) m,e 7 = (0) m n, e 8 = R (0), e 9 = R m n are the non-trivial ideals in R. Then the subgraph induced by Ω in AG(R) contains a subgraph isomorphic to the graph given in Fig 3.. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >. Hence n 3. Fig 3.: Forbidden subgraph for the projective plane Case. Suppose thatn = 3. LetJ be a non-trivial ideal inr such thatj m,m. Consider the set Ω = {f,...,f 9 } where f = m J, f = m m, f 3 = m m, f 4 = m (0), f 5 = (0) m, f 6 = (0) m, f 7 = (0) J, f 8 = R (0), f 9 = R m are the non-trivial ideals in R. Then the subgraph induced by Ω in AG(R) contains a subgraph isomorphic to the graph given in Fig 3.. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >. Let I be a non-trivial ideal in R such that I m. Consider the non-trivial ideals x = (0) m, x = m (0), x 3 = m m, y = (0) m, y = m m, y 3 = I (0), y 4 = I m, y 5 = I m in R. Then x iy j = (0) for every i,j and so K 3,5 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >. Suppose m and m, m are the only non-trivial ideals in R and R respectively. Then by Theorem., γ(ag(r)) = 0, a contradiction. Case. Assume that n =. Suppose there is an ideal I of R such that I m i for all i =,. Then by Proposition.4, R has at least three distinct non-trivial ideals J, J and J 3 such that J, J, J 3 m. Consider the non-trivial ideals a = m (0), a = (0) m, a 3 = m m, a 4 = (0) J, a 5 = m J, a 6 = (0) J, a 7 = m J in R. Then a i a j = (0) for every i j and so K 7 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >. IfR has at most two non-trivial ideals different fromm, then by Proposition.4 and Lemma.3, m is the only non-trivial ideal in R and if R has at most two non-trivial ideals different

7 Crosscap of the annihilating-ideal graph of a commutative ring 57 from m, then by Proposition.4 and Lemma.3, m is the only non-trivial ideal in R. Hence by Theorem.5, γ(ag(r)) = 0, a contradiction. Theorem 3.4. Let R = R R R n F F F m be a commutative ring with identity, where each (R i,m i ) is a local ring with m i {0} and each F j is a field and n,m. Let n i be the nilpotency of m i. Then γ(ag(r)) = if and only if R = R F and one of the following condition holds: (i) n = 3 andr has exactly 5 distinct non-trivial ideals, saym,m,i,i,i 3 withi i m (0) for every i =,, 3 and I i I j = (0) for somei j. (ii) n = 5 and R has exactly 4 distinct non-trivial ideals, saym,m,m3,m4. Proof. Assume that γ(ag(r))=. Suppose that n >. Consider the set Ω = {x,...,x 4, y,...,y 4 }, wherex = m n (0) (0) F (0) (0),x = (0) m n (0) F (0) (0),x 3 = (0) (0) (0) F (0) (0),x 4 = m n m n (0) F (0) (0),y = m (0) (0) (0),y = (0) m (0) (0), y 3 = m m (0) (0), y 4 = [R (0) (0),(0) R (0)]. Then the subgraph induced by Ω in AG(R) contains a subgraph isomorphic to the graph given in Fig 3.3. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >, a contradiction. Hence n = Fig 3.3: Forbidden subgraph for the projective plane Suppose that m >. Consider the non-trivial ideals a = (0) F (0) (0) (0), a = m n F (0) (0) (0),a 3 = m n (0) (0) (0),b = (0) (0) F (0) (0),b = (0) (0) (0) F 3 (0),b 3 = (0) (0) F F 3 (0), b 4 = m (0) F (0) (0),b 5 = m (0) (0) F 3 (0) inr. Thena i b j = (0) for every i,j and so K 3,5 is a subgraph ofag(r). By Lemma 3., γ(ag(r)) >, a contradiction. Hence m. Case. Assume that m =. Suppose that n >. Consider the set Ω = {e,...,e 9 }, where e = m n (0) F, e = m n F (0), e 3 = m (0) F, e 4 = m n (0) (0), e 5 = (0) F (0), e 6 = (0) F F, e 7 = (0) (0) F, e 8 = R (0) (0), e 9 = m (0) (0) are the non-trivial ideals in R. Then the subgraph induced by Ω in AG(R) contains a subgraph isomorphic to the graph given in Fig 3.. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >, a contradiction. Hence n =. Let I be any non-trivial ideal in R with I m. Consider the set Ω = {f,...,f 9 }, where f = I (0) F, f = I F (0), f 3 = m (0) F, f 4 = I (0) (0), f 5 = (0) F (0), f 6 = (0) F F, f 7 = (0) (0) F, f 8 = R (0) (0), f 9 = m (0) (0) are the non-trivial ideals inr. Then the subgraph induced by Ω inag(r) contains a subgraph isomorphic to the graph given in Fig 3.. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >, a contradiction. Hence m is the only non-trivial ideal in R and so by Theorem.5, γ(ag(r)) = 0, a contradiction. Case. Assume that m =. Let n > 5. Consider the set Ω = {x,...,x 4,y,...,y 4 }, where x = (0) F, x = m n F,x 3 = m n (0),x 4 = m n F,y = m 4 (0),y = m 3 (0),y 3 = m (0), y 4 = m (0) are the non-trivial ideals in R. Then the subgraph induced by Ω in AG(R) contain a subgraph isomorphic to the graph given in Fig 3.3. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >, a contradiction. Hence n 5. Subcase.. n =.

8 58 K. Selvakumar and P. Subbulakshmi Suppose there is an ideal I of R such that I m i for all i =,. Then by Proposition.4, R has at least three distinct non-trivial ideals I, I and I 3 such that I, I, I 3 m i for all i =,. Consider the non-trivial ideals u = (0) F, u = m F, u 3 = I F, u 4 = I F, v = m (0), v = I (0), v 3 = I (0),v 4 = I 3 (0) in R. Then u i v j = (0) for every i,j and so K 4,4 is a subgraph ofag(r). By Lemma 3.,γ(AG(R)) >, a contradiction. Hence by Proposition.4 and Lemma.3, m is the only non-trivial ideal in R and so by Theorem.5, γ(ag(r))=0, a contradiction. Subcase.. n = 3. Suppose there is an ideal I of R such that I m i for all i =,, 3. Then by Proposition.4, R has at least three distinct non-trivial ideals I, I and I 3 such that I, I, I 3 m i for all i =,. Suppose R has at least 4 non-trivial ideals different from m i for all i =,. Let I, I, I 3, I 4 be the distinct non-trivial ideals in R such that I i m,m for every i. Consider the non-trivial ideals u = (0) F, u = m F, u 3 = m (0), v = m (0), v = I (0), v 3 = I (0), v 4 = I 3 (0), v 5 = I 4 (0) in R. Then u i v j = (0) for every i,j and so K 3,5 is a subgraph ofag(r). By Lemma 3., γ(ag(r)) >, a contradiction. Hence R has exactly 3 non-trivial ideals different from m i for i =,. Hence m,m,i,i,i 3 are the only non-trivial ideals in R. SupposeI i m = (0) for somei. Consider the non-trivial idealsu = (0) F,u = m F, u 3 = m (0),u 4 = I i F,v = m (0),v = I (0),v 3 = I (0),v 4 = I 3 (0) inr. Then u i v j = (0) for every i,j and so K 4,4 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >, a contradiction. Hence I i m (0) for every i. Suppose I i is adjacent to I j for every j i. Let us assume that I I = (0), I I 3 = (0) and I I 3 = (0). Consider the set Ω = {u...u 9 }, whereu = m (0),u = I (0),u 3 = I 3 F, u 4 = (0) F, u 5 = I F, u 6 = I F, u 7 = m F, u 8 = I (0), u 9 = I 3 (0) are the non-trivial ideals in R. Then the subgraph induced by Ω in AG(R) contain a subgraph isomorphic to the graph given in Fig 3.4. and so by Theorem 6.5. [5, p.97], γ(ag(r)) >, a contradiction. Hence I I (0) or I I 3 (0) or I I 3 (0). m (0) I (0) I 3 F (0) F I F I F m F I (0) I 3 (0) Fig 3.4: Forbidden subgraph for the projective plane (0) F m F m (0) I (0) I F I 3 (0) m (0) m (0) I (0) m F I F I 3 F (0) F Fig 3.5: Projective embedding of AG(R F ) with n = 3, I i m (0) i,i I = (0),I I 3 = (0) and I I 3 (0)

9 Crosscap of the annihilating-ideal graph of a commutative ring 59 Subcase.3. n = 4. Suppose R has at least 3 non-trivial ideals different from m i for every i. Let J, J, J 3 be the distinct non-trivial ideals in R such that J i m,m,m3 for every i. Consider the non-trivial idealsu = (0) F,u = m 3 F,u 3 = m 3 (0),v = m (0),v = m (0),v 3 = J (0), v 4 = J (0), v 5 = J 3 (0) in R. Then u i v j = (0) for every i,j and so K 3,5 is a subgraph of AG(R). By Lemma 3., γ(ag(r)) >, a contradiction. Hence by Proposition.4 and Lemma.3, m, m, m3 are the only non-trivial ideals in R and so by Theorem.5, γ(ag(r)) = 0, a contradiction. Subcase.4. n = 5. Suppose R contains at least two distinct non-trivial ideals I, I such that I i m, m, m3, m 4 for i =,. Consider the non-trivial ideals c = (0) F, c = m 4 F, c 3 = m 4 (0), d = m 3 (0), d = m (0), d 3 = m (0), d 4 = I (0), d 5 = I (0) in R. Then c i d j = (0) for every i,j and so K 3,5 is a subgraph ofag(r). By Lemma 3., γ(ag(r)) >, a contradiction. Hence R contains at most one non-trivial ideal I such that I m, m, m3, m4. By Proposition.4,m,m,m3,m4 are the only non-trivial ideals in R. (0) F m 3 (0) m (0) m 4 (0) m 3 F m (0) m 4 F m F m 4 (0) m 3 (0) (0) F Fig 3.6: Projective embedding of AG(R F ) with n = 5 and m,m,m3,m4 are the only non-trivial ideals in R Converse follows from Fig. 3.5 and Fig References [] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, On the coloring of the annihilatingideal graph of a commutative ring, Discrete Math. 3, (0). [] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, Minimal prime ideals and cycles in annihilating-ideal graphs, Rocky Mountain J. Math. 43 (5), (03). [3] M. Afkhami, K. Khashyarmanesh, The cozero-divisor graph of a commutative ring, Southeast Asian Bull. Math. 35, (0). [4] D. F. Anderson, M. C. Axtell, J. A. Stickles, Zero-divisor graphs in commutative rings, Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana, S. E. Kabbaj, B. Olberding, I. Swanson), 3 45, Springer-Verlag, New York (0). [5] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 7, (999). [6] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 30, (008). [7] D. F. Anderson and A. Badawi, The total graph of a commutative ring without zero element, J. Algebra Appl. (8 pages) (0). [8] D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl. (8 pages) (03). [9] D. Archdeacon, Topological graph theory: a survey, Congr. Number. 5, 5 54 (996). [0] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, (969).

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