On the Non-Zero Divisor Graphs of Some Finite Commutative Rings

Abstract

The study of rings and graphs has been explored extensively by researchers. To gain a more effective understanding on the concepts of the rings and graphs, more researches on graphs of different types of rings are required. This manuscript provides a different study on the concepts of commutative rings and undirected graphs. The non-zero divisor graph, Γ(R) of a ring R is a simple undirected graph in which its set of vertices consists of all non-zero elements of R and two different vertices are joint by an edge if their product is not equal to zero. In this paper, the commutative rings are the ring of integers modulo n where n=8k and k≤3. The zero divisors are found first using the definition and then the non-zero divisor graphs are constructed. The manuscript explores some properties of non-zero divisor graph such as the chromatic number and the clique number. The result has shown that Γ(Z8k) is perfect.

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Malaysian Journal of Mathematical Sciences 17(2): 105–112 (2023)
https://doi.org/10.47836/mjms.17.2.02
Malaysian Journal of Mathematical Sciences
Journal homepage: https://mjms.upm.edu.my
On the Non-Zero Divisor Graphs of Some Finite Commutative Rings
Zai, N. A. F. O.1, Sarmin, N. H.∗1, Khasraw, S. M. S.2, Gambo, I.3, and Zaid, N.1
1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor
Bahru, Johor, Malaysia
2Department of Mathematics, College of Education, Salahaddin University-Erbil, Kurdistan Region, Iraq
3Department of Mathematical Sciences, Faculty of Science, Bauchi State University, Gadau, Nigeria
E-mail: nhs@utm.my
∗Corresponding author
Received: 1 June 2020
Accepted: 7 April 2023
Abstract
The study of rings and graphs has been explored extensively by researchers. To gain a more
effective understanding on the concepts of the rings and graphs, more researches on graphs of
different types of rings are required. This manuscript provides a different study on the concepts
of commutative rings and undirected graphs. The non-zero divisor graph, Γ(R)of a ring Ris a
simple undirected graph in which its set of vertices consists of all non-zero elements of Rand
two different vertices are joint by an edge if their product is not equal to zero. In this paper, the
commutative rings are the ring of integers modulo nwhere n= 8kand k≤3. The zero divisors
are found first using the definition and then the non-zero divisor graphs are constructed. The
manuscript explores some properties of non-zero divisor graph such as the chromatic number
and the clique number. The result has shown that Γ(
Z
8k)is perfect.
Keywords: commutative rings; non-zero divisor graphs; ring; zero divisors.

N. A. F. O. Zai et al. Malaysian J. Math. Sci. 17(2): 105–112(2023) 105 -112
1 Introduction
In abstract algebra, the theory of rings has its origin in the early 19th century when the commu-
tative and non-commutative rings are being explored. A ring is one of the fundamental algebraic
structures consisting of a set with two binary operations which are addition and multiplication
[5]. Several elementary results on the concept of rings and groups can be seen in Cohn [5], Aus-
lander and Buchsbaum [2], Rotman [11], Bourbaki [4], Mudaber et al. [6] and Romdhini et al.
[10].
In 1988, the zero-divisor graph was first introduced by Beck [3], where the main focus of the
research was on the graph coloring. A few decades later, the concept was further explored by
researchers by using different types of groups, rings and fields. Based on Beck’s work in [3],
Redmond [8] defined a simplified version of the zero-divisor graphs. The author only considered
the non-trivial zero divisors of the ring as the vertices of the graph instead of taking all non-zero
elements of the ring. In the following year, Redmond [9] explored the ideal-based zero divisor
graph of a commutative ring. This study reflects another structure of the subset of zero divisors
in a ring since total zero divisor graph engages both ring operations, namely addition and also
multiplication.
In 2020, Singh and Bhat [12] surveyed the most recent developments in describing the struc-
tural properties of zero-divisor graphs of finite commutative rings and their applications. The
idempotent graph of an abelian Rickart ring has been studied by Patil and Momale [7]. The au-
thors proved that the idempotent graph associated to zero divisor graph of an abelian Rickart ring
is weakly perfect.
This research is an extension to the concepts of graphs associated with rings. In this study, the
concepts of some commutative rings of order 8kare being extended to the graph theory, specifi-
cally to non-zero divisor graphs. The main focus is to construct the non-zero divisor graphs. The
manuscript begins by studying the concepts on rings and graphs. The ring of integers modulo
nwhere n= 8kand k≤3is considered. The zero divisors of the rings are found first. Then,
the non-zero divisor graphs are constructed. The graph-theoretic properties for the graph, which
are the clique number and chromatic number are analyzed and discussed. The perfectness of the
graph are investigated.
2 Preliminaries
This section is dedicated to present some fundamental definitions and properties that are re-
lated to the main topic. First, the formal definition of the zero divisors of a finite ring is provided
as follows.
Definition 2.1. Zero divisors of a ring are the elements that have product zero when multiplied with each
other. If pand qare two non-zero elements of a ring Rsuch that pq = 0, then pand qare divisors of 0.
Graphs associated to groups and rings have been studied extensively by researchers to study
the algebraic properties of the groups and rings. Some definitions related to graph theory are
presented in the following.
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N. A. F. O. Zai et al. Malaysian J. Math. Sci. 17(2): 105–112(2023) 105 -112
Definition 2.2. A finite graph, denoted as Γ, is an object with two sets, which are the edge set, E(Γ) and the
non-empty vertex set, V(Γ). The E(Γ) may be empty, but otherwise its elements are two-element subsets
of the vertex set.
The concepts of non-zero divisor graph is inspired by the zero divisor graph which is defined
as a simple graph with vertices of all zero divisors of a ring Rsuch that two different elements
pand qwill have an edge between them if and only if pq = 0, [1]. The authors studied the
interplay between the ring-theoretic properties of Rand the graph-theoretic properties of Γ(R).
The definition on the non-zero divisor graph is given as follows.
Definition 2.3. The non-zero divisor graph of a ring R, denoted by Γ(R), is a simple graph with its vertices
are all non-zero elements of a ring such that two distinct elements xand yare adjacent if and only if xy = 0.
In the final part of this manuscript, two graph properties namely the chromatic number and
clique number are determined based on the constructed non-zero divisor graph. Therefore, defi-
nitions of clique number and chromatic number are given as follows.
Definition 2.4. The minimum amount of colors needed to color the vertices of Γso that no two neighboring
vertices share the same color is the chromatic number and it is usually denoted by χ(Γ).
Definition 2.5. Clique is a complete subgraph in a graph, Γsub . Clique number is the greatest size of a
clique in an undirected graph, Γ. It is usually denoted by ω(Γ).
In addition, the graphs in this manuscript are found to be perfect. The definition of a perfect
graph is formally given as follows.
Definition 2.6. A perfect graph is a graph Γfor which every induced subgraph of Γhas chromatic number
equal to its clique number.
3 Results and Discussions
In this section, the zero divisors of the rings are found. The commutative rings that are being
considered in this study are the ring of integers modulo nwhere n= 8kand k≤3. The non-zero
divisor graphs are then constructed for each ring.
3.1 The zero divisors of ring of integers modulo 8k
The zero divisors are determined and calculated for the ring of integers modulo n= 8kwhere
k≤3by using Definition 2.1. The proposition and proofs are discussed as follows.
Proposition 3.1. Let
Z
8be the ring of integers modulo eight. The ring
Z
8has three zero divisors and 44
pairs of elements where the multiplication of these elements is not equal to zero.
Proof. The order of the ring
Z
8is eight since the elements of the ring are 0,1,2,3,4,5,6and 7. By
Definition 2.1, the zero divisors of
Z
8are determined. The zero divisors of
Z
8are 2,4and 6. There
are only three zero divisors found. The set consisting of pairs of elements that have product zero
is found and given as follows:
{(p, q)|pq = 0}={(4,2),(2,4),(4,4),(4,6),(6,4)}.
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N. A. F. O. Zai et al. Malaysian J. Math. Sci. 17(2): 105–112(2023) 105 -112
To find the multiplication of elements in the ring that is not equal to zero, the zero divisors
need to be excluded from the calculation. Since there are only seven non-zero elements of the
ring, the total pairs of elements are 7·7 = 49. Therefore, the total pairs of elements where the
multiplication of these pairs of those non-zero elements is not equal to zero is 49 −5(pairs of
elements from zero divisors) = 44.
This method of finding the zero divisors is repeated for the ring of integers modulo nwhere
n= 8kand k≤3. The zero divisors and the pairs of elements where the multiplication of the
non-zero elements is not equal to zero are given in the following table.
Table 1: Summary on the zero divisor of
Z
8kand its pairs of elements that do not have product zero.
Rings kOrder Zero Divisors pq = 0
Z
81 8 3 44
Z
16 2 16 7 208
Z
24 3 24 15 476
3.2 The construction of non-zero divisor graphs of ring of integers modulo 8k
The non-zero divisor graphs of the commutative rings are constructed based on the zero divi-
sors found in the previous subsection.
Proposition 3.2. The non-zero divisor graph of
Z
8,Γ(
Z
8)is an undirected graph with seven vertices and
19 edges.
Proof. Based on Definition 2.3,Γ(
Z
8)has seven vertices since
Z
8has seven non-zero elements. The
vertices of Γ(
Z
8)are connected when the multiplication of the elements is not equal to zero. As
stated in Table 1,
Z
8has 44 pairs of elements that do not have product zero. Note that the elements
x·x= 0 is not included as the edges of the non-zero divisor graph to prevent any loops. Hence,
Γ(
Z
8)has 19 undirected edges. The non-zero divisor graph, Γ(
Z
8)is shown in Figure 1.
Figure 1: The non-zero divisor graph of
Z
8.
It can be seen that the non-zero divisor graphs of
Z
8kare undirected graphs. Figure 2and
Figure 3illustrate the zero divisor graphs of
Z
8kwhere k= 2 and 3respectively.
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N. A. F. O. Zai et al. Malaysian J. Math. Sci. 17(2): 105–112(2023) 105 -112
Figure 2: The non-zero divisor graph of
Z
16. Figure 3: The non-zero divisor graph of
Z
24.
For the case of n=p, the non-zero divisor graph is a complete graph, Knsince there is no zero
divisor in the ring. The only divisors of Γ(
Z
p)are 1and p.
Theorem 3.1. Let Γ(
Z
p)be the non-zero divisor graph of ring of integers modulo pwhere pis a prime
number. Then, Γ(
Z
p)is isomorphic to Kp−1with exactly ((p−1)(p−2))/2edges.
Proof. Since
Z
phas no zero divisors, every vertex is connected with every other vertex. Therefore,
Γ(
Z
p)is isomorphic to Kp−1. As Knhas n(n−1)/2edges, then Kp−1has (p−1)(p−2)/2edges.
3.3 The properties of non-zero divisor graphs of ring of integers modulo 8k
In this subsection, two graph properties which are the chromatic number and the clique num-
ber are obtained after constructing the non-zero divisor graphs of ring of integers modulo 8k
where k≤3. The propositions on chromatic number and clique number of non-zero divisor
graph are provided.
Proposition 3.3. The chromatic number and the clique number of Γ(
Z
8)are six.
Proof. The chromatic number of Γ(
Z
8),χ(Γ(
Z
8)) is six since there are six colors that can be applied
on the vertices of Γ(
Z
8)so that two neighboring vertices do not share the similar color. For instance,
the vertex 1 and 3 have different colors of vertices since there is an edge between them. The non-
zero divisor graph of
Z
8with colors is shown in Figure 4. The set of maximum clique of this graph
is {1,2,3,5,6,7}. Therefore, based on Definition 2.6, the clique number of Γ(
Z
8),ω(Γ(
Z
8)) is also
six.
The method of finding the clique number and chromatic numbers is repeated to all the non-
zero divisor graphs constructed in subsection 3.2. Figure 5and Figure 6show the non-zero divisor
graphs of
Z
8kwith colors where k= 2 and 3. The clique number and chromatic number of Γ(
Z
16)
and Γ(
Z
24)are found to be 13 and 17 respectively.
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N. A. F. O. Zai et al. Malaysian J. Math. Sci. 17(2): 105–112(2023) 105 -112
Figure 4: The non-zero divisor graph of
Z
8with colors.
Figure 5: The non-zero divisor graph of
Z
16 with colors. Figure 6: The non-zero divisor graph of
Z
24 with colors.
Theorem 3.2. The non-zero divisor graph of the ring of integers modulo 8kwhere k≤3is perfect.
Proof. The clique number and the chromatic number are found first from the non-zero divisor
graph as in Proposition 3.3. It can be seen that the chromatic number and clique number of the
non-zero divisor graphs are equal. Therefore, by Definition 2.6, the non-zero divisor graphs of
Z
8kare perfect since ω(Γ(
Z
8k) = χ(Γ(
Z
8k)).
Based on Theorem 3.2, the general formula on the perfectness of the non-zero divisor graph of
commutative ring can be established.
Theorem 3.3. The non-zero divisor graph of a commutative ring is perfect.
Proof. Let Γ(
Z
n)be a non-zero divisor graph of a commutative ring of ring of integers modulo
nthat contains a clique of size qand a valid coloring using qcolors. Then, by finding the valid
coloring and a clique of size q, it can be seen that χ(Γ(
Z
n)) ≤qand ω(Γ(
Z
n)) ≥q. Note that
ω(Γ(
Z
n)) ≤χ(Γ(
Z
n)) since a new color is needed for each vertex in the largest clique. Therefore,
by combining the inequalities, q≤ω(Γ(
Z
n)≤χ(Γ(
Z
n)) ≤q. Since ω(Γ(
Z
n) = χ(Γ(
Z
n)) = q, by
Definition 2.6,Γ(
Z
n)is a perfect graph.
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4 Conclusions
This manuscript is being made to study the concepts of non-zero divisor graphs on different
types of rings. As the continuation of the researches on commutative rings, this study focuses to
determine the non-zero divisor graphs of commutative ring, which is the ring of integers modulo
nwhere n= 8kand k≤3. In conclusion, the zero divisors of
Z
8kare found and the non-zero
divisor graphs are constructed. The clique number and the chromatic number of the Γ(
Z
8k)are
determined and it is found that the non-zero divisor graph of ring of integers modulo nwhere
n= 8kand k≤3is perfect.
Acknowledgement The first author would like to acknowledge Universiti Teknologi Malaysia
(UTM) and Public Service Department (JPA) for the financial support through Program Pela-
jar Cemerlang (PPC). The authors would also like to thank the Ministry of Higher Education
Malaysia (MoHE) for the funding of this research through Fundamental Research Grant Scheme
(FRGS/1/2020/STG06/UTM/01/2) and UTM for the UTM Fundamental Research Grant (UTMFR)
Vote Number 20H70.
Conflicts of Interest The authors declare no conflict of interest.
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