On the graph of modules over commutative rings

Ansari-Toroghy, H.; Habibi, Sh.

doi:10.1216/rmj-2016-46-3-729

Cited by 2 publications

(2 citation statements)

References 11 publications

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“…Since G(τ * T ) is a tree, we have N(L) ∩ N(K) = ∅. By [5,Theorem 2.6], diam(G(τ * T )) ≤ 3. So every edge of G(τ * T ) is of the form {L, K}, {L, L i } or {K, K j }, for some i ∈ Λ and j ∈ Γ.…”

Section: Corollary 34 Assume Thatm Is a Multiplication Module Thenmentioning

confidence: 94%

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On the graph of modules over commutative rings II

Ansari-Toroghy¹,

Habibi²,

Hezarjaribi³

2018

Filomat

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Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(τ * T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1-19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(τ * T) is a bipartite graph. Also, we prove that if G(τ * T) is a tree, then G(τ * T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between χ(G(τ * T)) and ω(G(τ * T)).

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Section: Corollary 34 Assume Thatm Is a Multiplication Module Thenmentioning

confidence: 94%

“…There are many papers on assigning graphs to rings or modules (see, for example, [1, 4-7, 9, 16]). In [5], the present authors introduced and studied the graph G(τ * T ) (resp. AG(M)), called the quasi-Zariski topology-graph (resp.…”

Section: Introductionmentioning

confidence: 99%

On the graph of modules over commutative rings II

Ansari-Toroghy¹,

Habibi²,

Hezarjaribi³

2018

Filomat

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The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

Ansari-Toroghy

Habibi

2018

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let M be a module over a commutative ring and let Spec(M ) (resp. M ax(M )) be the collection of all prime (resp. maximal) submodules of M . We topologize Spec(M ) with Zariski topology, which is analogous to that for Spec(R), and consider M ax(M ) as the induced subspace topology. For any non-empty subset T of M ax(M ), we introduce a new graph G(τ m T ), called the Zariski topology-graph on the maximal spectrum of M . This graph helps us to study the algebraic (resp. topological) properties of M (resp. M ax(M )) by using the graph theoretical tools.1 2 HABIBOLLAH ANSARI-TOROGHY AND SHOKOOFEH HABIBI a new graph G(τ T ), called the Zariski topology-graph. This graph helps us to study algebraic (resp.topological) properties of modules (resp. Spec(M)) by using the graphs theoretical tools.a non-empty subset of Spec(M ) and distinct vertices N and L are adjacent if and only if V (N ) ∪ V (L) = T . There exists a topology on M ax(M ) having Z m (M ) = {V m (N ) : N ≤ M } as the set of closed sets of M ax(M ), where V m (N ) = {Q ∈ M ax(M ) : (Q : M ) ⊇ (N : M )}. We denote this topology by τ m M . In fact τ m M is the same as the subspace topology induced by τ M on M ax(M ).In this paper, we define a new graph G(τ m T ), called the Zariski topology-graph on the maximal spectrum of M, where T is a non-empty subset of M ax(M ), and by using this graph, we study algebraic (resp. topological) properties of M (resp.a non-empty subset of M ax(M ) and distinct vertices N and L are adjacent if and only if V m (N ) V m (L) = T . Let T be a non-empty subset of M ax(M ). As τ m M is the subspace topology induced by τ M on M ax(M ), one may think that G(τ T ) and G(τ m T ) have the identical nature. But the Example 3.5 (case (1)) shows that this is not true and these graphs are different. Also the Example 3.5 (case (2)) shows that for a non-empty subset T ′ of Spec(M ), under a condition, G(τ T ′ ) can be regarded as a subgraph of G(τ m T ), where T = T ′ ∩M ax(M ). Besides, this case denotes that G(τ T ′ ) is not a subgraph of G(τ m T ), necessarily. Moreover, it is shown that G(τ m T ) can not appear as a subgraph of G(τ T ′ ), where T ⊆ T ′ ⊆ Spec(M ), in general (see Example 3.5 (case (3)). So, the results related to G(τ m T ), where T is a non-empty subset of M ax(M ), do not go parallel to those of G(τ T ′ ), where T ′ is a non-empty subset of Spec(M ), necessarily. Based on the above remarks, it is worth to study Max-graphs separately. For any pair of submodules N ⊆ L of M and any element m of M , we denote L/N and the residue class of m modulo N in M/N by L and m respectively. For a submodule N of M , the colon ideal of M into N is defined by (N : M ) = {r ∈ R : rM ⊆ N } = Ann(M/N ). Further if I is an ideal of R, the submodule (N : M I) is defined by {m ∈ M : Im ⊆ N }. Moreover, Z (resp. Q) denotes the ring of integers (resp. the field of rational numbers). The prime radical √ N is defined to be the intersection of all prime submodules of M containing N , and in case N is not contained in any prime submodule, √ N is def...

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On the graph of modules over commutative rings

Cited by 2 publications

References 11 publications

On the graph of modules over commutative rings II

On the graph of modules over commutative rings II

The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

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