Domination number in the annihilating-submodule graph of modules over commutative rings
aa r X i v : . [ m a t h . A C ] J a n DOMINATION NUMBER IN THE ANNIHILATING-SUBMODULEGRAPH OF MODULES OVER COMMUTATIVE RINGS
H. ANSARI-TOROGHY AND S. HABIBI Department of pure Mathematics,Faculty of mathematical Sciences,University of Guilan, P. O. Box 41335-19141, Rasht, Iran.e-mail: [email protected] School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box:19395-5746, Tehran, Iran.Department of pure Mathematics, Faculty of mathematical Sciences, University of Guilan, P.O. Box 41335-19141, Rasht, Iran.e-mail: [email protected]
Abstract.
Let M be a module over a commutative ring R . The annihilating-submodule graph of M , denoted by AG ( M ), is a simple graph in which anon-zero submodule N of M is a vertex if and only if there exists a non-zeroproper submodule K of M such that NK = (0), where NK , the product of N and K , is denoted by ( N : M )( K : M ) M and two distinct vertices N and K are adjacent if and only if NK = (0). This graph is a submodule version ofthe annihilating-ideal graph and under some conditions, is isomorphic with aninduced subgraph of the Zariski topology-graph G ( τ T ) which was introducedin (The Zariski topology-graph of modules over commutative rings, Comm.Algebra., 42 (2014), 3283–3296). In this paper, we study the domination num-ber of AG ( M ) and some connections between the graph-theoretic propertiesof AG ( M ) and algebraic properties of module M . Introduction
Throughout this paper R is a commutative ring with a non-zero identity and M is a unital R -module. By N ≤ M (resp. N < M ) we mean that N is a submodule(resp. proper submodule) of M .Define ( N : R M ) or simply ( N : M ) = { r ∈ R | rM ⊆ N } for any N ≤ M .We denote ((0) : M ) by Ann R ( M ) or simply Ann ( M ). M is said to be faithful if Ann ( M ) = (0). Let N, K ≤ M . Then the product of N and K , denoted by N K , isdefined by ( N : M )( K : M ) M (see [6]). Define ann ( N ) or simply annN = { m ∈ M | m ( K : M ) = 0 } .The prime spectrum of M is the set of all prime submodules of M and denotedby Spec ( M ), M ax ( M ) is the set of all maximal submodules of M , and J ( M ), thejacobson radical of M , is the intersection of all elements of M ax ( M ), respectively. Mathematics Subject Classification.
Key words and phrases.
Commutative rings, annihilating-submodule graph, domination num-ber.This research was in part supported by a grant from IPM (No. 96130028).
There are many papers on assigning graphs to rings or modules (see, for example,[4, 7, 12, 13]). The annihilating-ideal graph AG ( R ) was introduced and studied in[13]. AG ( R ) is a graph whose vertices are ideals of R with nonzero annihilatorsand in which two vertices I and J are adjacent if and only if IJ = (0). Later, itwas modified and further studied by many authors (see [1, 2, 3, 18, 20]).In [7], the present authors introduced and studied the graph G ( τ T ) (resp. AG ( M )),called the Zariski topology-graph (resp. the annihilating-submodule graph ), where T is a non-empty subset of Spec ( M ). AG ( M ) is an undirected graph with vertices V ( AG ( M ))= { N ≤ M | there exists(0) = K < M with
N K = (0) } . In this graph, distinct vertices N, L ∈ V ( AG ( M ))are adjacent if and only if N L = (0) (see [9, 10]). Let AG ( M ) ∗ be the subgraphof AG ( M ) with vertices V ( AG ( M ) ∗ ) = { N < M with ( N : M ) = Ann ( M ) | thereexists a submodule K < M with ( K : M ) = Ann ( M ) and N K = (0) } . By [7,Theorem 3.4], one conclude that AG ( M ) ∗ is a connected subgraph. Note that M isa vertex of AG ( M ) if and only if there exists a nonzero proper submodule N of M with ( N : M ) = Ann ( M ) if and only if every nonzero submodule of M is a vertexof AG ( M ). Clearly, if M is not a vertex of AG ( M ), then AG ( M ) = AG ( M ) ∗ . In[8, Lemma 2.8], we showed that under some conditions, AG ( M ) is isomorphic withan induced subgraph of the Zariski topology-graph G ( τ T ).In this paper, we study the domination number of AG ( M ) and some connec-tions between the graph-theoretic properties of AG ( M ) and algebraic properties ofmodule M .A prime submodule of M is a submodule P = M such that whenever re ∈ P forsome r ∈ R and e ∈ M , we have r ∈ ( P : M ) or e ∈ P [17].The notations Z ( R ) and N il ( R ) will denote the set of all zero-divisors, the set ofall nilpotent elements of R , respectively. Also, Z R ( M ) or simply Z ( M ), the set ofzero divisors on M , is the set { r ∈ R | rm = 0 for some 0 = m ∈ M } . If Z ( M ) = 0,then we say that M is a domain. An ideal I ≤ R is said to be nil if I consist ofnilpotent elements.Let us introduce some graphical notions and denotations that are used in whatfollows: A graph G is an ordered triple ( V ( G ) , E ( G ) , ψ G ) consisting of a nonemptyset of vertices, V ( G ), a set E ( G ) of edges, and an incident function ψ G that asso-ciates an unordered pair of distinct vertices with each edge. The edge e joins x and y if ψ G ( e ) = { x, y } , and we say x and y are adjacent. The number of edges incidentat x in G is called the degree of the vertex x in G and is denoted by d G ( v ) or simply d ( v ). A path in graph G is a finite sequence of vertices { x , x , . . . , x n } , where x i − and x i are adjacent for each 1 ≤ i ≤ n and we denote x i − − x i for existing an edgebetween x i − and x i . The distance between two vertices x and y , denoted d ( x, y ),is the length of the shortest path from x to y . The diameter of a connected graph G is the maximum distance between two distinct vertices of G . For any vertex x of a connected graph G , the eccentricity of x , denoted e ( x ), is the maximum ofthe distances from x to the other vertices of G . The set of vertices with minimumeccentricity is called the center of the graph G , and this minimum eccentricity valueis the radius of G . For some U ⊆ V ( G ), we denote by N ( U ), the set of all verticesof G \ U adjacent to at least one vertex of U and N [ U ] = N ( U ) ∪ { U } .A graph H is a subgraph of G , if V ( H ) ⊆ V ( G ), E ( H ) ⊆ E ( G ), and ψ H isthe restriction of ψ G to E ( H ). A subgraph H of G is a spanning subgraph of G if OMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH 3 V ( H ) = V ( G ). A spanning subgraph H of G is called a perfect matching of G ifevery vertex of G has degree 1.A clique of a graph is a complete subgraph and the supremum of the sizes ofcliques in G , denoted by cl ( G ), is called the clique number of G . Let χ ( G ) denote thechromatic number of the graph G , that is, the minimal number of colors neededto color the vertices of G so that no two adjacent vertices have the same color.Obviously χ ( G ) ≥ cl ( G ).A subset D of V ( G ) is called a dominating set if every vertex of G is either in D or adjacent to at least one vertex in D . The domination number of G , denotedby γ ( G ), is the number of vertices in a smallest dominating set of G . A totaldominating set of a graph G is a set S of vertices of G such that every vertexis adjacent to a vertex in S . The total domination number of G , denoted by γ t ( G ), is the minimum cardinality of a total dominating set. A dominating setof cardinality γ ( G ) ( γ t ( G )) is called a γ -set ( γ t -set). A dominating set D is aconnected dominating set if the subgraph < D > induced by D is a connectedsubgraph of G . The connected domination number of G , denoted by γ c ( G ), is theminimum cardinality of a connected dominating set of G . A dominating set D isa clique dominating set if the subgraph < D > induced by D is complete in G .The clique domination number γ cl ( G ) of G equals the minimum cardinality of aclique dominating set of G . A dominating set D is a paired-dominating set if thesubgraph < D > induced by D has a perfect matching. The paired-dominationnumber γ pr ( G ) of G equals the minimum cardinality of a paired-dominating set of G . A vertex u is a neighbor of v in G , if uv is an edge of G , and u = v . The setof all neighbors of v is the open neighborhood of v or the neighbor set of v , and isdenoted by N ( v ); the set N [ v ] = N ( v ) ∪ { v } is the closed neighborhood of v in G .Let S be a dominating set of a graph G , and u ∈ S . The private neighborhoodof u relative to S in G is the set of vertices which are in the closed neighborhoodof u , but not in the closed neighborhood of any vertex in S \ { u } . Thus theprivate neighborhood P N ( u, S ) of u with respect to S is given by P N ( u, S ) = N [ u ] \ ( ∪ v ∈ S \{ u } N [ v ]). A set S ⊆ V ( G ) is called irredundant if every vertex v of S has at least one private neighbor. An irredundant set S is a maximal irredundantset if for every vertex u ∈ V \ S , the set S ∪{ u } is not irredundant. The irredundancenumber ir ( G ) is the minimum cardinality of maximal irredundant sets. There areso many domination parameters in the literature and for more details one can refer[15].A bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V ; that is, U and V are each independent sets and complete bipartite graph on n and m vertices,denoted by K n,m , where V and U are of size n and m , respectively, and E ( G )connects every vertex in V with all vertices in U . Note that a graph K ,m is calleda star graph and the vertex in the singleton partition is called the center of thegraph. We denote by P n a path of order n (see [14]).In section 2, a dominating set of AG ( M ) is constructed using elements of thecenter when M is an Artinian module. Also we prove that the domination numberof AG ( M ) is equal to the number of factors in the Artinian decomposition of M and we also find several domination parameters of AG ( M ). In section 3, we studythe domination number of the annihilating-submodule graphs for reduced rings H. ANSARI-TOROGHY AND S. HABIBI with finitely many minimal primes and faithful modules. Also, some relationsbetween the domination numbers and the total domination numbers of annihilating-submodule graphs are studied.The following results are useful for further reference in this paper.
Proposition 1.1.
Suppose that e is an idempotent element of R . We have thefollowing statements.(a) R = R × R , where R = eR and R = (1 − e ) R .(b) M = M × M , where M = eM and M = (1 − e ) M .(c) For every submodule N of M , N = N × N such that N is an R -submodule M , N is an R -submodule M , and ( N : R M ) = ( N : R M ) × ( N : R M ).(d) For submodules N and K of M , N K = N K × N K such that N = N × N and K = K × K .(e) Prime submodules of M are P × M and M × Q , where P and Q are primesubmodules of M and M , respectively. Proof.
This is clear. (cid:3)
We need the following results.
Lemma 1.2. (See [5, Proposition 7.6].) Let R , R , . . . , R n be non-zero ideals of R . Then the following statements are equivalent:(a) R = R × . . . × R n ;(b) As an abelian group R is the direct sum of R , . . . , R n ;(c) There exist pairwise orthogonal idempotents e , . . . , e n with 1 = e + . . . + e n , and R i = Re i , i = 1 , . . . , n . Lemma 1.3. (See [16, Theorem 21.28].) Let I be a nil ideal in R and u ∈ R besuch that u + I is an idempotent in R/I . Then there exists an idempotent e in uR such that e − u ∈ I . Lemma 1.4. (See [9, Lemma 2.4].) Let N be a minimal submodule of M and let Ann ( M ) be a nil ideal. Then we have N = (0) or N = eM for some idempotent e ∈ R . Proposition 1.5.
Let
R/Ann ( M ) be an Artinian ring and let M be a finitelygenerated module. Then every nonzero proper submodule N of M is a vertex in AG ( M ). Theorem 1.6. (See [9, Theorem 2.5] .) Let
Ann ( M ) be a nil ideal. There existsa vertex of AG ( M ) which is adjacent to every other vertex if and only if M = eM ⊕ (1 − e ) M , where eM is a simple module and (1 − e ) M is a prime modulefor some idempotent e ∈ R , or Z ( M ) = Ann (( N : M ) M ) , where N is a nonzeroproper submodule of M or M is a vertex of AG ( M ) . Theorem 1.7. (See [9, Theorem 3.3] .) Let M be a faithful module. Then thefollowing statements are equivalent. (a) χ ( AG ( M ) ∗ ) = 2 . (b) AG ( M ) ∗ is a bipartite graph with two nonempty parts. (c) AG ( M ) ∗ is a complete bipartite graph with two nonempty parts. (d) Either R is a reduced ring with exactly two minimal prime ideals, or AG ( M ) ∗ is a star graph with more than one vertex. OMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH 5
Corollary 1.8. (See [9, Corollary 3.5].) Let R be a reduced ring and assume that M is a faithful module. Then the following statements are equivalent.(a) χ ( AG ( M ) ∗ ) = 2.(b) AG ( M ) ∗ is a bipartite graph with two nonempty parts.(c) AG ( M ) ∗ is a complete bipartite graph with two nonempty parts.(d) R has exactly two minimal prime ideals. Proposition 1.9. (See [15, Proposition 3.9].) Every minimal dominating set in agraph G is a maximal irredundant set of G .2. Domination number in the annihilating-submodule graph forArtinian modules
The main goal in this section, is to obtain the value certain domination param-eters of the annihilating-submodule graph for Artinian modules.Recall that M is a vertex of AG ( M ) if and only if there exists a nonzero propersubmodule N of M with ( N : M ) = Ann ( M ) if and only if every nonzero submoduleof M is a vertex of AG ( M ). In this case, the vertex N is adjacent to every othervertex. Hence γ ( AG ( M )) = 1 = γ t (( AG ( M ))). So we assume that throughoutthis paper M is not a vertex of AG ( M ). Clearly, if M is not a vertex of AG ( M ), then AG ( M ) = AG ( M ) ∗ . We start with the following remark which completely characterizes all modulesfor which γ (( AG ( M ))) = 1. Remark 2.1.
Let
Ann ( M ) be a nil ideal. By Theorem 1.6, there exists a vertex of AG ( M ) which is adjacent to every other vertex if and only if M = eM ⊕ (1 − e ) M ,where eM is a simple module and (1 − e ) M is a prime module for some idempotent e ∈ R , or Z ( M ) = Ann (( N : M ) M ), where N is a nonzero proper submodule of M or M is a vertex of AG ( M ). Now, let Ann ( M ) be a nil ideal and M be a domainmodule. Then γ (( AG ( M ))) = 1 if and only if M = eM ⊕ (1 − e ) M , where eM is asimple module and (1 − e ) M is a prime module for some idempotent e ∈ R . Theorem 2.2.
Let M be a f.g Artinian local module. Assume that N is the uniquemaximal submodule of M . Then the radius of AG ( M ) is or and the center of AG ( M ) is { K ⊆ ann ( N ) | K = (0) is a submodule in M } .Proof. If N is the only non-zero proper submodule of M , then AG ( M ) ∼ = K , e ( N ) = 0 and the radius of AG ( M ) is 0. Assume that the number of non-zeroproper submodules of M is greater than 1. Since M is f.g Artinian module, thereexists m ∈ N , m > N m = (0) and N m − = (0). For any non-zerosubmodule K of M , KN m − ⊆ N N m − = (0) and so d ( N m − , K ) = 1. Hence e ( N m − ) = 1 and so the radius of AG ( M ) is 1. Suppose K and L are arbitrarynon-zero submodules of M and K ⊆ ann ( N ). Then KL ⊆ KN = (0) and hence e ( K ) = 1. Suppose (0) = K ′ * ann ( N ). Then K ′ N = (0) and so e ( K ′ ) > AG ( M ) is { K ⊆ ann ( N ) | K = (0) is a submodule in M } . (cid:3) Corollary 2.3.
Let M be a f.g Artinian local module and N is the unique maximalsubmodule of M . Then the following hold good.(a) γ ( AG ( M )) = 1.(b) D is a γ -set of AG ( M ) if and only if D ⊆ ann ( N ). H. ANSARI-TOROGHY AND S. HABIBI
Proof. ( a ) Trivial from Theorem 2.6.( b ) Let D = { K } be a γ -set of AG ( M ). Suppose K * ann ( N ). Then KN = (0) andso N is not dominated by K , a contradiction. Conversely, suppose D ⊆ ann ( N ).Let K be an arbitrary vertex in AG ( M ). Then KL ⊆ N L = (0) for every L ∈ D .i.e., every vertex K is adjacent to every L ∈ D . If | D | >
1, then D \ { L ′ } is also adominating set of AG ( M ) for some L ′ ∈ D and so D is not minimal. Thus | D | = 1and so D is a γ -set by ( a ). (cid:3) Theorem 2.4.
Let M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all ≤ i ≤ n and n ≥ . Then the radius of AG ( M ) is and the center of AG ( M ) is { K ⊆ J ( M ) | K = (0) is a submodule in M } .Proof. Let M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all 1 ≤ i ≤ n and n ≥
2. Let J i be the unique maximal submodule in M i with nilpotency n i .Note that M ax ( M ) = { N , . . . , N n | N i = M ⊕ . . . ⊕ M i − ⊕ J i ⊕ M i +1 ⊕ . . . ⊕ M n , ≤ i ≤ n } is the set of all maximal submodules in M . Consider D i = (0) ⊕ . . . ⊕ (0) ⊕ J n i − i ⊕ (0) ⊕ . . . ⊕ (0) for 1 ≤ i ≤ n . Note that J ( M ) = J ⊕ . . . ⊕ J n is the Jacobsonradical of M and any non-zero submodule in M is adjacent to D i for some i . Let K be any non-zero submodule of M . Then K = ⊕ ni =1 K i , where K i is a submoduleof M i . Case 1 . If K = N i for some i , then KD j = (0) and KN j = (0) for all j = i . Notethat N ( K ) = { (0) ⊕ . . . ⊕ (0) ⊕ L i ⊕ (0) ⊕ . . . ⊕ (0) | J i L i = (0), L i is a nonzerosubmodule in M i } . Clearly N ( K ) ∩ N ( N j ) = (0), d ( K, N j ) = 2 for all j = i , andso K − D i − D j − N j is a path in AG ( M ). Therefore e ( K ) = 3 and so e ( N ) = 3for all N ∈ M ax ( M ). Case 2 . If K = D i and K i ⊆ J i for all i . Then KD i = (0) for all i . Let L be anynon-zero submodule of M with KL = (0). Then LD j = (0) for some j , K − D j − L is a path in AG ( M ) and so e ( K ) = 2. Case 3 . If K i = M i for some i , then KD i = (0), KN i = (0) and KD j = (0) forsome j = i . Thus K − D j − D i − N i is a path in AG ( M ), d ( K, N i ) = 3 and so e ( K ) = 3. Thus e ( K ) = 2 for all K ⊆ J ( M ). Further note that in all the casescenter of AG ( M ) is { K ⊆ J ( M ) | K = (0) is a submodule in M } . (cid:3) In view of Theorems 2.2 and 2.4, we have the following corollary.
Corollary 2.5.
Let M = ⊕ ni =1 M i , where M i is a simple module for all 1 ≤ i ≤ n and n ≥
2. Then the radius of AG ( M ) is 1 or 2 and the center of AG ( M ) is ∪ ni =1 D i ,where D i = (0) ⊕ . . . ⊕ (0) ⊕ M i ⊕ (0) ⊕ . . . ⊕ (0) for 1 ≤ i ≤ n . Theorem 2.6.
Let M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all ≤ i ≤ n and n ≥ . Then γ ( AG ( M )) = n .Proof. Let N i be the unique maximal submodule in M i with nilpotency n i . LetΩ = { D , D , . . . , D n } , where D i = (0) ⊕ . . . ⊕ (0) ⊕ J n i − i ⊕ (0) ⊕ . . . ⊕ (0) for 1 ≤ i ≤ n . Note that any non-zero submodule in M is adjacent to D i for some i . Therefore N [Ω] = V ( AG ( M )), Ω is a dominating set of AG ( M ) and so γ ( AG ( M )) ≤ n .Suppose S is a dominating set of AG ( M ) with | S | < n . Then there exists N ∈ M ax ( M ) such that N K = (0) for all K ∈ S , a contradiction. Hence γ ( AG ( M )) = n . (cid:3) In view of Theorem 2.6, we have the following corollary.
OMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH 7
Corollary 2.7.
Let M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all1 ≤ i ≤ n and n ≥
2. Then(a) ir ( AG ( M )) = n .(b) γ c ( AG ( M )) = n .(c) γ t ( AG ( M )) = n .(d) γ cl ( AG ( M )) = n .(e) γ pr ( AG ( M )) = n , if n is even and γ pr ( AG ( M )) = n + 1, if n is odd. Proof.
Consider the γ -set of AG ( M ) identified in the proof of Theorem 2.6. ByProposition 1.9, Ω is a maximal irredundant set with minimum cardinality andso ir ( AG ( M )) = n . Clearly < Ω > is a complete subgraph of AG ( M ). Hence γ c ( AG ( M )) = γ t ( AG ( M )) = γ cl ( AG ( M )) = n . If n is even, then < Ω > has a per-fect matching and so Ω is a paired dominating set of AG ( M ). Thus pr ( AG ( M )) = n . If n is odd, then < Ω ∪ K > has a perfect matching for some K ∈ V ( AG ( M )) \ Ω.and so Ω ∪ K is a paired dominating set of AG ( M ). Thus γ pr ( AG ( M )) = n if n even and γ pr ( AG ( M )) = n + 1 if n is odd. (cid:3) Let M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all 1 ≤ i ≤ n and n ≥
2. Then by Theorem 2.4, radius of AG ( M ) is 2. Further, by Theorem2.6, the domination number of AG ( M ) is equal to n , where n is the number ofdistinct maximal submodules of M . However, this need not be true if the radiusof AG ( M ) is 1. For, consider the ring M = M ⊕ M , where M and M aresimple modules. Then AG ( M ) is a star graph and so has radius 1, whereas M has two distinct maximal submodules. The following corollary shows that a moreprecise relationship between the domination number of AG ( M ) and the number ofmaximal submodules in M , when M is finite. Corollary 2.8.
Let M be a finite module and γ (( AG ( M ))) = n . Then either M = M ⊕ M , where M , M are simple modules or M has n maximal submodules. Proof.
When γ (( AG ( M ))) = 1, proof follows from [9, Corollary 2.12]. When γ (( AG ( M ))) = n , then M cannot be M = M ⊕ M , where M , M are sim-ple modules. Hence M = ⊕ mi =1 M i , where M i is a f.g Artinian local module for all1 ≤ i ≤ m and m ≥
2. By Theorem 2.6, γ (( AG ( M ))) = m . Hence by assumption m = n . i.e., M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all 1 ≤ i ≤ n and n ≥
2. One can see now that M has n maximal submodules. (cid:3) The relationship between γ t (( AG ( M ))) and γ (( AG ( M )))The main goal in this section is to study the relation between γ t (( AG ( M ))) and γ (( AG ( M ))). Theorem 3.1.
Let M be a module. Then γ t (( AG ( M ))) = γ (( AG ( M ))) or γ t (( AG ( M ))) = γ (( AG ( M ))) + 1 .Proof. Let γ t (( AG ( M ))) = γ (( AG ( M ))) and D be a γ -set of AG ( M ). If γ (( AG ( M ))) =1, then it is clear that γ t (( AG ( M ))) = 2. So let γ (( AG ( M ))) > k = M ax { n | there exist L , . . . , L n ∈ D ; ⊓ ni =1 L i = 0 } . Since γ t (( AG ( M ))) = γ (( AG ( M ))), we have k ≥
2. Let L , . . . , L k ∈ D be such that ⊓ ki =1 L i = 0.Then S = {⊓ ki =1 L i , annL , . . . , annL k } ∪ D \ { L , . . . , L k } is a γ t -set. Hence γ t (( AG ( M ))) = γ (( AG ( M ))) + 1. (cid:3) H. ANSARI-TOROGHY AND S. HABIBI
In the following result we find the total domination number of AG ( M ). Theorem 3.2.
Let S be the set of all maximal elements of the set V ( AG ( M )) . If | S | > , then γ t (( AG ( M ))) = | S | .Proof. Let S be the set of all maximal elements of the set V ( AG ( M )), K ∈ S and | S | >
1. First we show that K = ann ( annK ) and there exists m ∈ M suchthat K = ann ( m ). Let K ∈ S . Then annK = 0 and so there exists 0 = m ∈ annK . Hence K ⊆ ann ( annK ) ⊆ ann ( m ). Thus by the maximality of K , we have K = ann ( annK ) = ann ( m ). By Zorn’ Lemma it is clear that if V ( AG ( M )) = ∅ ,then S = ∅ . For any K ∈ S choose m K ∈ M such that K = ann ( m K ). Weassert that D = { Rm K | K ∈ S } is a total dominating set of AG ( M ). Since forevery L ∈ V ( AG ( M )) there exists K ∈ S such that L ⊆ K = ann ( m K ), L and Rm K are adjacent. Also for each pair K, K ′ ∈ S , we have ( Rm K )( Rm K ′ ) = 0.Namely, if there exists m ∈ ( Rm K )( Rm K ′ ) \ { } , then K = K ′ = ann ( m ). Thus γ t (( AG ( M ))) ≤ | S | . To complete the proof, we show that each element of anarbitrary γ t -set of AG ( M ) is adjacent to exactly one element of S . Assume to thecontrary, that a vertex L ′ of a γ t -set of AG ( M ) is adjacent to K and K ′ , for K, K ′ ∈ S . Thus K = K ′ = annL ′ , which is impossible. Therefore γ t (( AG ( M ))) = | S | . (cid:3) Theorem 3.3.
Let R be a reduced ring, M is a faithful module, and | M in ( R ) | < ∞ .If γ (( AG ( M ))) > , then γ t (( AG ( M ))) = γ (( AG ( M ))) = | M in ( R ) | .Proof. Since R is reduced, M is a faithful module, and γ (( AG ( M ))) >
1, wehave | M in ( R ) | >
1. Suppose that
M in ( R ) = { p , . . . , p n } . If n = 2, the re-sult follows from Corollary 1.8. Therefore, suppose that n ≥
3. Define d p i M = p . . . p i − p i +1 . . . p n M , for every i = 1 , . . . , n . Clearly, d p i M = 0, for every i =1 , . . . , n . Since R is reduced, we deduce that d p i M p i M = 0. Therefore, every p i M is a vertex of AG ( M ). If K is a vertex of AG ( M ), then by [11, Corollary 3.5],( K : M ) ⊆ Z ( R ) = ∪ ni =1 p i . It follows from the Prime Avoidance Theorem that( K : M ) ⊆ p i , for some i , 1 ≤ i ≤ n . Thus p i M is a maximal element of V ( AG ( M )),for every i = 1 , . . . , n . From Theorem 3.2, γ t (( AG ( M ))) = | M in ( R ) | . Now, weshow that γ (( AG ( M ))) = n . Assume to the contrary, that B = { J , . . . , J n − } is a dominating set for AG ( M ). Since n ≥
3, the submodules p i M and p j M ,for i = j are not adjacent (from p i p j = 0 ⊆ p k it would follow that p i ⊆ p k ,or p j ⊆ p k which is not true). Because of that, we may assume that for some k < n − J i = p i M for i = 1 , . . . , k , but none of the other of submodulesfrom B are equal to some p s M (if B = { p M, . . . , p n − M } , then p n M wouldbe adjacent to some p i M , for i = n ). So, every submodule in { p k +1 M, ..., p n M } is adjacent to a submodule in { J k +1 , ..., J n − } . It follows that for some s = t ,there is an l such that ( p s M ) J l = 0 = ( p t M ) J l . Since p s * p t , it follows that J l ⊆ p t M , so J l = 0, which is impossible, since the ring R is reduced. So γ t (( AG ( M ))) = γ (( AG ( M ))) = | M in ( R ) | . (cid:3) Theorem 3.3 leads to the following corollary.
Corollary 3.4.
Let R be a reduced ring, M is a faithful module, and | M in ( R ) | < ∞ , then the following are equivalent.(a) γ ( AG ( M )) = 2.(b) AG ( M ) is a bipartite graph with two nonempty parts.(c) AG ( M ) is a complete bipartite graph with two nonempty parts. OMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH 9 (d) R has exactly two minimal primes. Proof.
Follows from Theorem 3.3 and Corollary 1.8. (cid:3)
In the following theorem the domination number of bipartite annihilating-submodulegraphs is given.
Theorem 3.5.
Let M be a faithful module. If AG ( M ) is a bipartite graph, then γ (( AG ( M ))) ≤ .Proof. Let M be a faithful module. If AG ( M ) is a bipartite graph, then fromTheorem 1.7, either R is a reduced ring with exactly two minimal prime ideals,or AG ( M ) is a star graph with more than one vertex. If R is a reduced ringwith exactly two minimal prime ideals, then the result follows by Corollary 3.4. If AG ( M ) is a star graph with more than one vertex, then we are done. (cid:3) The next theorem is on the total domination number of the annihilating-submodulegraphs of Artinian modules.
Theorem 3.6.
Let M = ⊕ ni =1 M i , where M i is a f.g Artinian local module for all ≤ i ≤ n , n ≥ , and M = M ⊕ M , where M , M are simple modules. Then γ t (( AG ( M ))) = γ (( AG ( M ))) = | M in ( R ) | .Proof. By Proposition 1.5, every nonzero proper submodule of M is a vertex in AG ( M ). So, the set of maximal elements of V ( AG ( M )) and M ax ( M ) are equal.Let M = ⊕ ni =1 M i , where ( M i , J i ) is a f.g Artinian local module for all 1 ≤ i ≤ n and n ≥
2. Let
M ax ( M ) = { N i = M ⊕ . . . ⊕ M i − ⊕ J i ⊕ M i +1 ⊕ . . . ⊕ M n | ≤ i ≤ n } . By Theorem 3.2, γ t (( AG ( M ))) = | M ax ( M ) | . In the sequel, we prove that γ (( AG ( M ))) = n . Assume to the contrary, the set { K , . . . , K n − } is a dominatingset for AG ( M ). Since M = M ⊕ M , where M , M are simple modules, we findthat K i N s = K i N t = 0, for some i, t, s , where 1 ≤ i ≤ n − ≤ t, s ≤ n . Thismeans that K i = 0, a contradiction. (cid:3) The following theorem provides an upper bound for the domination number ofthe annihilating-submodule graph of a Noetherian module.
Theorem 3.7.
If R is a Notherian ring and M a f.g module, then γ (( AG ( M ))) ≤| Ass ( M ) | < ∞ .Proof. By [19], Since R is a Notherian ring and M a f.g module, | Ass ( M ) | < ∞ . Let Ass ( M ) = { p , ..., p n } where p i = ann ( m i ) for some m i ∈ M for every i = 1 , . . . , n .Set A = { Rm i | ≤ i ≤ n } . We show that A is a dominating set of AG ( M ). Clearly,every Rm i is a vertex of AG ( M ), for i = 1 , . . . , n (( p i M )( m i R ) = 0). If K is avertex of AG ( M ), then [19, Corollary 9.36] implies that ( K : M ) ⊆ Z ( M ) = ∪ ni =1 p i .It follows from the Prime Avoidance Theorem that ( K : M ) ⊆ p i , for some i ,1 ≤ i ≤ n . Thus K ( Rm i ) = 0, as desired. (cid:3) The remaining result of this paper provides the domination number of theannihilating-submodule graph of a finite direct product of modules.
Theorem 3.8.
For a module M , which is a product of two ( nonzero ) modules, oneof the following holds: (a) If M ∼ = F × D , where F is a simple module and D is a prime module, then γ ( AG ( M )) = 1 . (b) If M ∼ = D × D , where D and D are prime modules which are not simple,then γ ( AG ( M )) = 2 . (c) If M ∼ = M × D , where M is a module which is not prime and D is aprime module, then γ ( AG ( M )) = γ ( AG ( M )) + 1 . (d) If M ∼ = M × M , where M and M are two modules which are not prime,then γ ( AG ( M )) = γ ( AG ( M )) + γ ( AG ( M )) .Proof. Parts ( a ) and ( b ) are trivial.( c ) With no loss of generality, one can assume that γ ( AG ( M )) < ∞ . Supposethat γ ( AG ( M )) = n and { K , . . . , K n } is a minimal dominating set of AG ( M ).It is not hard to see that { K × , . . . , K n × , × D } is the smallest dominatingset of AG ( M ).( d ) We may assume that γ ( AG ( M )) = m and γ ( AG ( M )) = n , for some positiveintegers m and n . Let { K , . . . , K m } and { L , . . . , L n } be two minimal dominatingsets in AG ( M ) and AG ( M ), respectively. It is easy to see that { K × , . . . , K m × , × L . . . × L n } is the smallest dominating set in AG ( M ). (cid:3) References [1] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr, and F. Shaveisi,
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