# Prime Decomposition of Zero Divisor Graph in a Commutative Ring

Authors: A. Kuppan; J. Ravi Sankar
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

LetR be a commutative ring and let ΓZn be the zero divisor graph of a commutative ring R, whose vertices are nonzero zero divisors of Zn, and such that the two vertices u,v are adjacent if n divides uv. In this paper, we introduce the concept of prime decomposition of zero divisor graph in a commutative ring and also discuss some special cases of ΓZ3p, ΓZ5p, ΓZ7p, and ΓZpq.


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## Body

## 1. Introduction

Let us consider the finite, simple, and undirected graphs to discuss about the prime decomposition in the form of zero divisor graphs [1].Here, complete graph withn vertices is represented by Kn and the complete bipartite graph is represented by Km,n. Also, a path of length k is by Sk. (ie) Sk=K1,k. Let K1,miβ be the star graph of β copies where m be the prime number with ′i′ tupes. Let L be the family of all subgraph and L is an edge disjoint decomposition with αi copies of Hi, where i∈1,2,…,r and α is a positive integer. Furthermore, if each Hii∈1,2,3,…,r is isomorphic to a graph H, then we say that G has an H− decomposition.The zero divisor graphs play an important role in algebraic properties and algebraic structures such as a commutative ring. The concept of a zero divisor graph is a commutative ring was proposed by Beck’s [2]. The general terminology and notation everything based on the papers [[3–7]]. In this paper, we investigate the prime decomposition of ΓZpq into K1,mi star graph with mi edges and obtain the following results. We already investigated the concept of decomposition of zero divisor graph for some special cases ΓZn where n=pq,p2,p2q [8].LetR be the ring and GR be the graph of the ring. If two distinct vertices r.s∈GR are adjacent then we can represent it as rs=0. Beck’s initiated his work with a chromatic number of the graph. In this paper, we discuss the prime decomposition of Zpq into K1,mi star graph with mi edges.


## 2. Preliminaries

Definition 1.
(see [4]). Let R be a commutative ring (with 1) and let Z (R) be its set of zero-divisors. We associate a (simple) graph ΓR to R with vertices ZR∗=ZR−0, the set of nonzero zero-divisor of R, and for distinct x,y∈ZR∗ the vertices x and y are adjacent if and only if xy=0. Thus, ΓR is the empty graph if and only if R is an integral domain.Definition 2.
A graphG is decomposable into H1,H2,H3,…,Hk if G has subgraphs H1,H2,H3,…,Hk such that(1)
Each edge ofG belongs to one of the Hi′s for some i=1,2,3,...,k(2)
Ifi≠j, then Hi and Hj have no edges in common.


## 3. Prime Decomposition of Zero Divisor Graph

Theorem 1.
Ifp is any prime number, p>3 and r≥0 then the graph ΓZ3p admits a K1,mi2p−1/mi,K1,r2− prime decomposition if and only if p−1≡rmodmi where mi is any prime less than p.Proof 1.
Suppose thatp is any prime number, p>3 and r≥0, we have, ΓZ3p be the nonzero zero divisor graph. The vertex set of ΓZ3p is 3,6,9,…,3p−1,p,2p.Case 1.
Let us considerp−1|mi. If the graph ΓZ3p is prime decomposition into 2⌊p−1/mi⌋ copies of K1,mi then there exists p−1≡rmodmi where mi is any prime number less than p.Case 2.
Let us considerp−1|/mi then their exists a remainder r. If the graph ΓZ3p is prime decomposition into 2⌊p−1/mi⌋ copies of K1,mi and (copies of K1,r≅r copies of K1,2) then there exists p−1≡rmodmi where mi is any prime number less than p. Above-given case 1 and case 2 clearly show that the prime decomposition of ΓZ3p into K1,mi2p−1/mi,K1,r2 then p−1≡rmodmi.
Conversely, suppose thatp−1≡rmodmi where m is any prime less than p.
Let us considerp=5 and m=2,3. If 4≡0mod2 then there exists 4 copies of K1,2. If 4≡1mod3 then there exists 2 copies of K1,3 and (2 copies of K1,1≅ one copy of K1,2).
Let us takep=7 and m=2,3,5. If 6≡0mod2 then there exists 6 copies of K1,2. If 6≡0mod3 then there exists 4 copies of K1,3. If 6≡1mod5 then there exists 2 copies of K1,5 and (2 copies of K1,1≅ one copy of K1,2).
In general, takep is any prime number p>3 and mi is prime numbers less than p. Clearly, If p−1≡rmodmi then ΓZ3p is a prime decomposition into 2⌊p−1/mi⌋ copies of K1,m and (2 copies of K1,r≅r copies of K1,2). Hence the proof (see Figure 1).Figure 1
ΓZ15.Example 1.
Let us takep=7 and q=11 the graph ΓZ15 as the example of Theorem 1.Theorem 2.
Ifpis any prime number,p>5andr≥0then the graphΓZ5padmits aK1,mi4⌊p−1/mi⌋,K1,r4−prime decomposition if and only ifp−1≡rmodmiwheremiis any prime less than p.Proof 2.
Suppose thatp is any prime number, p>5, we have, ΓZ5p be the nonzero zero divisor graph with isomorphic to K4,p−1. The vertex set of ΓZ5p is 5,10,15,…,5p−1,p,2p,3p,4p.Case 3.
Let us takep−1|mi. If the prime decomposition of ΓZ5p into 4⌊p−1/mi⌋ copies of K1,mi then there exists p−1≡rmodmi where mi is any prime numbers less than p.Case 4.
Let us takep−1|/mi. If the prime decomposition of ΓZ5p into 4⌊p−1/mi⌋ copies of K1,mi. Clearly, the remaining edges are (r copies of K1,4≅ 4 copies of K1,r) where r is the remainder of p−1,mi. Then, there exists p−1≡rmodmi. Clearly, shows that the above cases prime decomposition of ΓZ5p into K1,mi4⌊p−1/mi⌋,K1,r4 then p−1≡rmodmi.
Conversely, suppose thatp−1≡rmodmi where m is any prime numbers less than p.
Let us takep=7 and mi=2,3,5. If 6≡0mod2 then 12 copies of K1,2. If 6≡0mod3 then 8 copies of K1,3. If 6≡1mod5 then 4 copies of K1,5 and (4 copies of K1,1≅ one copy of K1,4).
Let us takep=11 and mi=2,3,5,7. If 10≡0mod2 then 20 copies of K1,2. If 10≡1mod3 then 12 copies of K1,3 and (4 copies of K1,1≅ one copy of K1,4. If 10≡0mod5 then 8 copies of K1,5. If 10≡3mod7 then 4 copies of K1,7 and (4 copies of K1,3≅ 3 copies of K1,4).
In general, takep is any prime number p>5 and mi is prime numbers less than p. Clearly, If p−1≡rmodmi then ΓZ5p is a prime decomposition into 4⌊p−1/mi⌋ copies of K1,mi and (4 copies of K1,r≅r copies of K1,4). Hence the proof (see Figure 2).Figure 2
ΓZ35.Example 2.
Let us takep=7 and q=11 the graph ΓZ15 as the example of Theorem 2Theorem 3.
Ifpis any prime number,p>7andr≥0then the graphΓZ7padmits aK1,mi6⌊p−1/mi⌋,K1,r6−prime decomposition if and only ifp−1≡rmodmiwheremiis any prime less than p.Proof 3.
Suppose thatp is any prime number, p>7, we have, ΓZ7p be the nonzero zero divisor graph with isomorphic to K6,p−1. The vertex set of ΓZ6p is 7,14,21,…,7p−1,p,2p,3p,4p,5p,6p.Case 5.
Let us takep−1|mi the prime decomposition of ΓZ7p into 6⌊p−1/mi⌋ copies of K1,mi then there exists p−1≡rmodmi where mi is any prime numbers less than p.Case 6.
Let us takep−1|/mi the prime decomposition of ΓZ7p into 6⌊p−1/mi⌋ copies of K1,mi. Clearly, the remaining edges are (r copies of K1,6≅ 6 copies of K1,r) where r is the remainder of p−1,mi. Then, there exists p−1≡rmodmi. Clearly, shows that the above-given cases prime decomposition of ΓZ7p into K1,mi6⌊p−1/mi⌋,K1,r6 then p−1≡rmodmi.
Conversely, suppose thatp−1≡rmodmi where mi is any prime numbers less than p. Let us take p=11 and mi=2,3,5,7. If 10≡0mod2 then 30 copies of K1,2. If 10≡1mod3 then 18 copies of K1,3 and (6 copies of K1,1≅ one copy of K1,6). If 10≡0mod5 then 12 copies of K1,5. If 10≡3mod7 then 6 copies of K1,7 and (6 copies of K1,3≅ 3 copies of K1,6). Let us take p=13 and mi=2,3,5,7,11. If 12≡0mod2 then 36 copies of K1,2. If 12≡0mod3 then 24 copies of K1,3. If 12≡2mod5 then 12 copies of K1,5 and (6 copies of K1,2≅ 2 copies of K1,6). If 12≡5mod7 then 6 copies of K1,7 and (6 copies of K1,5≅ 5 copies of K1,6). If 12≡1mod11 then 6 copies of K1,11 and (6 copies of K1,1≅ one copy of K1,6). In general, take p is any prime number p>7 and mi is prime numbers less than p. Clearly, If p−1≡rmodmi then ΓZ7p is a prime decomposition into 6⌊p−1/mi⌋ copies of K1,mi and 6 copies of K1,r≅r copies of K1,6. Hence the proof (see Figure 3).Figure 3
ΓZ77.Example 3.
Let us takep=7 and q=11 the graph ΓZ15 as the example of Theorem 3,Theorem 4.
Ifpandqare any distinct prime numbers,q>pandr≥0then the graphΓZpqadmits aK1,mip−1⌊q−1/mi⌋,K1,rp−1−prime decomposition if and only ifq−1≡rmodmiwheremiis any prime less than p.Proof 4.
Suppose thatp and q are any distinct prime numbers, q>p, we have, ΓZpq be the nonzero zero divisor graph with isomorphic to Kp−1,q−1. The vertex set of ΓZpq is p,2p,3p,…,pq−1,q,2q,3q,…,qp−1.Case 7.
Let us takeq−1|mi. If prime decomposition of ΓZpq into p−1⌊q−1/mi⌋ copies of K1,mi then there exists q−1≡rmodmi where mi is any prime numbers less than p.Case 8.
Let us takeq−1|/mi the prime decomposition of ΓZpq into p−1⌊q−1/mi⌋ copies of K1,mi. Clearly, the remaining edges are (r copies of K1,q−1≅q−1 copies of K1,r) where r is the remainder of q−1,mi. Then, there exists q−1≡rmodmi. Clearly, shows that the above cases prime decomposition of ΓZpq into K1,mip−1⌊q−1/mi⌋,K1,rp−1 then q−1≡rmodmi.
Conversely, suppose thatq−1≡rmodmi where mi is any prime numbers less than p.
Let us takep and q are any distinct prime numbers and mi is any prime less than q. Clearly, the above-given theorems show. If q−1≡rmodmi then ΓZpq is a prime decomposition into p−1⌊q−1/mi⌋ copies of K1,mi and p−1 copies of K1,r≅r copies of K1,p−1. Hence the proof.


## 4. Conclusion

In this paper, we have defined the Prime Decomposition of the Zero Divisor Graph of a commutative ring. Also, some special cases ofΓZ3p, ΓZ5p, ΓZ7p, and ΓZpq are established. In the future, we will study some more properties and applications of Prime Decomposition of Zero Divisor Graph. [9].



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