# Residual Division Graph of Lattice Modules

Authors: Ganesh Gandal; R. Mary Jothi; Narayan Phadatare
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetL be a multiplicative lattice and M be a lattice module over L. In this paper, we assign a graph to M called residual division graph RG(M) in which the element N∈M is a vertex if there exists 0M≠P∈M such that NP=0M and two vertices N1,N2 are adjacent if N1N2=0M (where N1N2=N1:IMN2:IMIM). It is proved that such a graph with the greatest element IM which does not belong to the vertex set is nonempty if and only if M is a prime lattice module. Also, we provide conditions such that RGM is isomorphic to a subgraph of Zariski topology graph ĢXM with respect to X.


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## Body

## 1. Introduction

In our everyday life, we found that numerous issues are dealt with the assistance of graphs. Extraordinarily, the idea of the coloring of graphs assumes a significant function in computer sciences. To examine the coloring of rings, I. Beck first presented the zero-divisor graphs of a commutative ring with unity (see [1]). This examination of the coloring of a commutative ring was further studied by Anderson and Naseer (see [2]).The ring structure is firmly associated with ideals more than elements and so it has the right to present a graph with vertices as ideals instead of elements. In this direction, M. Behboodi et al. studied the annihilating-ideal graphAGR with vertex set VAGR contain all those ideals of ring R whose annihilators are nonzero(see [3, 4]). Thereafter, Ansari-Toroghy et al. expanded this work for R-module M, where R is a commutative ring. They studied the algebraic as well as topological properties of M with the help of annihilating-submodule graph and the Zariski topology graph (see [5]). Recently, the idea of the zero divisor has likewise been applied in Boolean algebra, poset, and lattices (see [6, 7, 78]).A complete latticeL=L,∨,∧,0L,1L is a multiplicative if there is a defined binary operation called multiplication, denoted by “·”, which is commutative and associative such that the greatest element 1L works as the multiplicative identity and for an arbitrary index set Ia.∨α∈Ibα=∨α∈Ia.bα, where a,bα∈L and 0L is the least element of L. It is interesting to note that the purpose behind the development of multiplicative lattice is to generalize lattices of ring ideals (see [9, 10]). Also, it is observed that the annihilating-ideal graph of a commutative ring R with unity has close ties with a multiplicative lattice of ideals of R. As far as the study of Johnson [11] is concerned, a lattice module is just an extension of a multiplicative lattice. It becomes worthy to study the graph AGR over a commutative ring R with unity with the help of a lattice module M over L.Definition 1 (see [12]).
Alattice moduleM over the multiplicative lattice L is a complete lattice with least element 0M and greatest element IM if a multiplication between elements of L and M, represented by xN∈M, where x∈L and N∈M, which satisfies the following properties:(1)
xyN=xyN(2)
∨αxα∨βNβ=∨αβxαNβ(3)
1LN=N(4)
0LN=0M, for all x,y,xα∈L and for all N,Nβ∈M
Note that, forP,Q∈M and x∈L, we define P:Q=∨a∈L∣aQ≤P and P:x=∨K∈M|xK≤P. Here, the operation ″:″ is called residual division (see [12]). Also, note that A,C∈M with A≤C; the interval B∈M|A≤B≤C which is denoted by C/A is a lattice module over a multiplicative lattice L with the multiplication a⋅B=aB∨A, where a∈L (see [12]).
Furthermore, for more definitions and concept of lattice modules and multiplicative lattice, see [9–19].
The semicomplement graphΓM of lattice module M introduced and investigated by Phadatare et al. (see [20]). In the recent paper [5], Ansari-Toroghy and Habibi have highlighted the closed sets in Zariski topology on prime spectrum SpecM of R-module M and defined new graph called Zariski topology graphGτT, where T⊆SpecM which is nonempty. They studied the relationship between GτT and annihilating-submodule graph AGM/∩T (see [5]). Thereafter, this graph GτT generalized to lattice modules M over a C-lattice L and the Zariski topology graphĢXM was studied (see [21]).
Throughout the paper,M denotes a lattice module over a multiplicative lattice L, and for N,K∈M, we define NK=N:IMK:IMIM.
The aim of this paper is to generalize the annihilating-submodule graph of a module to the lattice moduleM over a multiplicative lattice L and introduce the residual division graphRGM whose vertex set is N∈M|thereexists0M≠K∈MsuchthatNK=0M and in RGM; two vertices A,B are adjacent if and only if AB=0M. Apart from this, we will investigate interrelationship between ĢXM and RGM/∧X, where ∧X is the meet of all elements in X.
Throughout the paper,M denotes a lattice module over a multiplicative lattice L.


## 2. Some Graph Theoretic Notions

We consider only undirected graphs. Thus, we adopt the notationG=V,E, where V=VG is the set of vertices and E=EG is the set of edges of G. A graph G is an empty if VG=∅. The number of edges incident on a vertex v is called a degree of vertex v and it is denoted by dv. A graph G is said to be k-regular if the degree of each vertex in G is k. Distance between the vertices a and b is the length of the shortest path between them, which is denoted by da,b. Consider da,b=∞ if there is no path between a and b. diamG=supda,c|a,c∈VG is the diameter of a graph G. Length of shortest cycle in G is called the girth of G, denoted by grG. A clique of graph G is its maximal complete subgraph, and the minimum number of cliques required to cover all the vertices of graph G is called the partition number, denoted by θG. In a graph G, a subset S⊆VG is supposed to be independent if no two vertices in S are adjacent. The size of maximum independent set in a graph G called as independence number, denoted by αG. For a vertex x∈VG, ΓGx=y|xy∈EG denotes the set of all neighbors of x′ in G. A graph G is said to be perfect if θH=αH, for every induced subgraph H of G. Strongly perfect and very strongly perfect graphs are the classes of the perfect graph.For further information, the reader may refer [22, 23].


## 3. Residual Division GraphRGM

Definition 2.
Theresidual division graph RGM of M is a graph with vertices VRGM=N∈Mthereexists0M≠K∈MsuchthatNK=0M, where distinct vertices P and Q are adjacent if and only if PQ=0M.Example 1.
Figure1 represents the residual division graph RGM of M with the vertex set VRGM=0M,A,B,C,Q, where M represents lattice module over L (see Figure 2 and 3).
We essentially need the following two Lemmas throughout this article.Figure 1
Multiplicative latticeL.Figure 2
Lattice moduleM over L.Figure 3
RGM.Lemma 1 (see [12]).
Forx∈L and P,Q,R∈M, the following holds:(1)
IfP≤Q, then P:R≤Q:R(2)
Q∧R:P=Q:P∧R:P(3)
x≤xP:P(4)
xP:x≤P(5)
P:QQ≤P(6)
IfP≤Q, then R:Q≤R:PLemma 2 (see [12]).
ForN∈M, N:IMIM:IM=N:IM.The following lemma gives a condition under which a nonzero proper element of lattice moduleM is a vertex of RGM.Lemma 3.
Let0M≠N be a proper element of M. Then, N∈VRGM if 0M:N≠0M:IM or 0M:N:IM≠0M.Proof.
Suppose that0M≠N is a proper element of M and 0M:N≠0M:IM. Then, 0M:NIM≠0M. Now, let K=0M:NIM. Therefore, NK=N:IM0M:NIM:IMIM≤N:IM0M:NIM=0M:NN:IMIM≤0M:NN=0M. This implies that NK=0M; consequently, N∈VRGM.Callialp and Tekir [14] introduced the notion of multiplication lattice modules.Definition 3 (see [14]).
A lattice moduleM is said to be multiplication lattice module if for each K∈M there exists an element x∈L such that K=xIM.Lemma 4 (see [14]).
A lattice moduleM is a multiplication lattice module if and only if K=K:IMIM, for all K∈M.Lemma 5.
Let0M≠N be a proper element of multiplication lattice module M. Then, N∈VRGM if and only if 0M:N:IM≠0M.Proof.
Suppose that0M≠N∈M is a vertex of RGM. Then, by definition, there exists 0M≠K∈M such that NK=0M. Therefore, N:IMK:IMIM=0M. Since M is multiplication, by Lemma 4, K:IMIM=K; therefore, N:IMK:IMIM=N:IMK=0M. This implies 0M:N:IM=K. However, K≠0M; therefore, 0M:N:IM≠0M. Converse part follows from Lemma 3.Co-multiplication lattice module is introduced and characterized by F. Callialp et al. (see [16]).Definition 4 (see [16]).
Lattice moduleM is called a co-multiplication lattice module if for each P∈M, there exists an element x∈L such that P=0M:x.
The following characterization plays an important role in the study of residual division graphRGM.Lemma 6 (see [16]).
Lattice moduleM is a co-multiplication if and only if K=0M:0M:K for every element K∈M.The following theorem is the immediate consequence of Lemma6.Theorem 1.
Every proper element0M≠N of a co-multiplication lattice module M is a vertex of RGM.Proof.
Let0M≠N is a proper element of co-multiplication lattice module M. By Lemma 3, to prove N∈VRGM, we have to show that 0M:N≠0M:IM or 0M:N:IM≠0M. Suppose that 0M:N=0M:IM. Then, by Lemma 1(6), 0M:0M:N=0M:0M:IM. Since M is a co-multiplication lattice module over a C-lattice L, by Lemma 6, N=0M:0M:N=0M:0M:IM=IM which is contradiction to N<IM; consequently, 0M:N≠0M:IM.In the above three results, we studied various conditions on lattice moduleM under which proper elements 0M≠N∈M becomes a vertex. However, then the natural question arises: can the greatest element IM be a vertex in residual division graph RGM?The following lemma answers the above question.Lemma 7.
Greatest elementIM of M is a vertex if and only if there exists a proper element 0M≠N∈M such that N:IM=0M:IM.Proof.
Suppose that greatest elementIM of a lattice module M is a vertex. Then, there exists a 0M≠P∈M such that PIM=0M. Therefore, P:IMIM:IMIM=0M. However, IM:IMIM=IM; therefore, P:IMIM=0M, and hence, P:IM=0M:IM. Conversely, suppose that N:IM=0M:IM, where N is a proper element of M with 0M≠N. Then, NIM=N:IMIM:IMIM=N:IMIM because of IM:IMIM=IM. Since N:IM=0M:IM, we have NIM=N:IMIM=0M:IMIM≤0M. By Lemma 1(5), 0M:IMIM≤0M; therefore, NIM=0M; hence, the greatest element IM is a vertex.Theorem 2.
Let the greatest elementIM of M is not a vertex of RGM. Then, 0M≠N∈M is a vertex if and only if 0M:N:IM≠0M.Proof.
Suppose that0M≠N∈M is in VRGM. Then, there exists 0M≠P∈M such that NP=0M. Therefore, NP=N:IMP:IMIM=0M and so P:IMIM≤0M:N:IM. Since IM is not a vertex, by Lemma 7, P:IMIM≠0M, and hence, 0M:N:IM≠0M. Converse follows from Lemma 3.In [13], Al-Khouja studied the relationship between the maximal (prime) elements of lattice module M and the maximal (prime) elements of multiplicative lattice L. “If N is a prime element of a lattice module M over a multiplicative lattice L, then N:IM is a prime element of multiplicative lattice L (see [13]).”According to Callialp et al. (see [15]), a lattice module M over a multiplicative lattice L is prime if the least element 0M is prime element of M.The following characterization is done by Callialp et al. (see [15]).Lemma 8 (see [15]).
Least element0M of M is a prime if and only if 0M:IM=0M:N, for all 0M≠N∈M.Lemma8 helps us to characterize the prime lattice module.Theorem 3.
Let the greatest elementIM of M is not a vertex of RGM. Then, RGM=∅ if and only if M is a prime lattice module.Proof.
Suppose thatRGM=∅ and M is not a prime lattice module over L. Then, least element 0M is not a prime element of M. Therefore, by Lemma 8, there exists proper element 0M≠N∈M such that 0M:IM≠0M:N, and hence, by Lemma 3, N is in VRGM, contradiction to RGM=∅; consequently, M is prime. Conversely, M is a prime, and there exists N∈M such that N∈VRGM. Then, by definition, there exists 0M≠KinM such that NK=0M. This implies that N:IMK:IMIM=0M, so N:IMK:IM=0M:IM. Since 0M is a prime element of M, 0M:IM prime element of L. Therefore, N:IM=0M:IM or K:IM=0M:IM. This follows by Lemma 7 that IM is a vertex, a contradiction. Consequently, RGM=∅.Theorem 4.
For givenM, RGM is connected and diamRGM≤3.Proof.
Suppose thatP,Q∈VRGM such that P≠Q. If PQ=0M, by definition, we have a path P−Q of length one. Now, suppose that PQ≠0M.Case (1): ifP2=PP=0M and Q2=QQ=0M, then PPQ=P:IMP:IMQ:IMIM=0M and PQQ=P:IMQ:IMQ:IMIM=0M. This implies that P is adjacent to PQ and PQ is adjacent to Q, i.e., P−PQ−Q is a path of length equal to 2.Case (2): ifP2=0M and Q2≠0M, since Q∈VRGM, there exists 0M≠Q1∈M such that QQ1=0M. If PQ1=0M, then we have a path P−Q1−Q with dP,Q=2. Suppose that PQ1≠0M. Then, PPQ1=P2Q1=0M, and therefore, P−PQ1−Q is a path of length 2 because QQ1=0M. Similarly, if Q2=0M and P2≠0M, then we have a path with length 2.Case (3): ifPQ≠0M, P2≠0M, and Q2≠0M, by definition, there exist 0M≠N,K such that PN=0M=QK. If N=K, then P−N=K−Q is a path of length 2. Now, suppose that N≠K and NK=0M. Since PN=0M=QK, we have a path P−N−K−Q of length 3, i.e., dP,Q=3. Above cases implies that dP,Q≤3; consequently, diamRGM≤3.Corollary 1.
IfRGM contain a cycle, then grRGM≤4.In [21], Phadatare et al. introduced the quasi-prime element of M.Definition 5 (see [21]).
A proper elementN∈M is said to be quasi-prime if N:IM is a quasi-prime element of L.
The following lemma follows from Theorem3.Lemma 9.
Let the greatest elementIM of M is not a vertex of RGM. IfRGM=∅, then the least element 0M is a quasi-prime element of M.SpecqM is a collection of all quasi-prime elements of M and for N∈M, set DN=K∈SpecqM|N:IM≤K:IM (see [21]).We basically need the following lemma.Lemma 10 (see [21]).
LetM be a lattice module and Y⊆SpecqM. Then, Y⊆DN if and only if N:IM≤∧Y:IM.Phadatare et al. [21] employed closed set DN to introduce the Zariski topology graph ĢXM with respect to X⊆SpecqM.Definition 6.
ForX⊆SpecqM, we define an undirected graph ĢXM associated with X, called Zasiski topology graph with respect to X with vertex set VĢXM=N∈M|thereexists0M≠K∈MsuchthatDN∪DK=XandDN≠X,DK≠X and distinct vertices N and K are adjacent if and only if DN∪DK=X.Remark 1.
By Proposition 3.1 of [21] and Lemma 2, for N,K∈M, DNK=DN:IMK:IMIM=DN∪DK.
Note that, forY⊆SpecqM, an interval K∈M:∧Y≤K≤M denoted by M/∧Y=M¯ is a lattice module over a multiplicative lattice L with the multiplication a⋅K=aK∨∧Y, where a∈L.Theorem 5.
LetM be a lattice module, and M/∧Y=IM¯ is not a vertex of RGM/∧Y. Then, RGM/∧Y is isomorphic to subgraph of ĢYM.Proof.
Suppose thatN¯=N/∧Y∈VRGM/∧Y. By definition, there exists ∧Y≠K/∧Y∈M/∧Y such that N/∧Y is adjacent to K/∧Y. Therefore, N¯K¯=N/∧YK/∧Y=N/∧Y:IM/∧YK/∧Y:IM/∧YIM/∧Y=∧Y. Note that N/∧Y:IM/∧YK/∧Y:IM/∧Y=N:IMK:IMIM∨∧Y/∧Y. Therefore, we have N¯K¯=N/∧YK/∧Y=N:IMK:IMIM∨∧Y/∧Y=∧Y. This impies that NK=N:IMK:IMIM≤∧Y; therefore, by Lemma 1(1), NK:IM≤∧Y:IM, and hence, by Lemma 10, DNK=Y. Since DNK=DN∪DK, we have Y=DN∪DK. If DN=Y, then N:IM=∧X:IM; therefore, by Lemma 7, IM¯ is a vertex of RGM/∧Y which is a contradiction; consequently, DN≠Y. In the same line, we have DK≠Y. Thus, N∈VĢYM such that N is adjacent to K.Corollary 2 (see [21]).
ForX⊆SpecqM,ĢXM is nonempty if and only X=D∧Xand∧X∉SpecqM.The following is the characterization of nonempty residual division graphRGM/∧Y.Theorem 6.
IfM/∧Y=IM¯ is not a vertex of RGM/∧Y, then RGM/∧Y=∅ if and only if ĢYM=∅.Proof.
Suppose thatIM¯ is not a vertex of RGM/∧Y and RGM/∧Y=∅. Then, by Lemma 9, ∧Y is a quasi-prime element of M; therefore, by Corollary 2, ĢYM=∅. Converse follows from Theorem 6.Theorem 7.
The greatest elementM/∧Y=IM¯ is a vertex in RGM/∧Y if there exists N<IM such that ∧Y<N and DN=Y.Proof.
Suppose there existsN<IM such that ∧Y<N and DN=Y. To prove that IM¯∈VRGM/∧Y, it suffices to prove that there exists ∧Y≠K¯ such that K¯IM¯=∧Y. Note that ∧Y<N<IM. By definition, NIM¯=N/∧YIM/∧Y=N/∧Y:IM/∧YIM/∧Y:IM/∧YIM/∧Y=N:IMIM∨∧Y/∧Y. Since DN=Y, we have, for all P∈Y, N:IM≤P:IM; therefore, N:IM≤∧P∈YP:IM, and hence, by Lemma 1(2), N:IM≤∧P∈YP:IM. This implies that N:IM≤∧Y:IM, i.e., N:IMIM≤∧Y; therefore, N:IMIM∨∧Y/∧Y=∧Y; consequently, IM¯ is a vertex in RGM/∧Y.
ForN<IM, N represents q-radical of N, and it is defined as N=∧DN. If N=N, then N is said to be q-radical element of M.
“If the natural mapψ:SpecqM⟶SpecqL/0M:IM defined by ψN=N:IM¯ surjective, then N:IM=N:IM (see [24]).”Theorem 8.
If the natural mapψ is surjective and A,B are adjacent vertices in ĢYM. Then, A/∧Y and B/∧Y are adjacent in RGM/∧Y.Proof.
To show thatA/∧Y and B/∧Y are adjacent in RGM/∧Y, we have to prove that A:IMIM/∧Y≠∧Y and B:IMIM/∧Y≠∧Y with A:IMIM/∧YB:IMIM/∧Y=∧Y. Suppose that A,B are adjacent vertices in ĢYM. Then, we have DA∪DB=Y with DA,DB≠Y. However, DA∪DB=DA:IMB:IMIM; therefore, Y=DA:IMB:IMIM, and hence, ∧Y=∧DA:IMB:IMIM=A:IMB:IMIM≤A:IMIM∩B:IMIM. Thus, ∧Y≤A:IMIM∩B:IMIM. Also, note that A:IMIM/∧YB:IMIM/∧Y=A:IMIM/∧Y:IM/∧YB:IMIM/∧Y:IM/∧YIM/∧Y=A:IMIM:IMB:IMIM:IMIM∨∧Y/∧Y. Since ψ is surjective and DN=DN:IMIM, we have A:IMIM:IM=A:IM=A:IM; therefore, A:IMIM/∧YB:IMIM/∧Y=A:IMIM:IMB:IMIM:IMIM∨∧Y/∧Y=A:IMB:IMIM∨IM/∧Y. However, A:IMB:IMIM≤A:IMB:IMIM≤A:IMB:IMIM=AB=∧Y; therefore, A:IMIM/∧YB:IMIM/∧Y=A:IMB:IMIM∨IM/∧Y=∧Y. Now, it remains to prove that A:IMIM/∧Y≠∧Y and A:IMIM/∧Y≠∧Y. If A:IMIM/∧Y=∧Y, then A:IMIM=∧Y. Therefore, A:IMIM≤A:IMIM=∧Y, and hence, A:IM≤∧Y:IM. By Lemma 10, DN=Y, a contradiction. Consequently, A:IMIM/∧Y≠∧Y. Similarly, we have B:IMIM/∧Y≠∧Y.


## 4. Conclusion

In this paper, we introduced the residual division graph of the lattice module and characterized the co-multiplication lattice module and prime lattice module. Also, we characterized vertex set of residual division graph of the multiplication lattice module. We found that the residual division graph of an interval lattice module is isomorphic with the Zariski topology graph.



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