Abstract
Let R be a commutative ring and let be the zero divisor graph of a commutative ring , whose vertices are nonzero zero divisors of , and such that the two vertices are adjacent if divides . 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 , , , and .
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 with vertices is represented by and the complete bipartite graph is represented by . Also, a path of length is by . (ie) . Let be the star graph of copies where be the prime number with tupes. Let be the family of all subgraph and is an edge disjoint decomposition with copies of , where and is a positive integer. Furthermore, if each is isomorphic to a graph , then we say that has an 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 into star graph with edges and obtain the following results. We already investigated the concept of decomposition of zero divisor graph for some special cases where [8].
Let be the ring and be the graph of the ring. If two distinct vertices are adjacent then we can represent it as . Beck’s initiated his work with a chromatic number of the graph. In this paper, we discuss the prime decomposition of into star graph with 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 to R with vertices , the set of nonzero zero-divisor of R, and for distinct the vertices x and y are adjacent if and only if . Thus, is the empty graph if and only if is an integral domain.
Definition 2. A graph is decomposable into if has subgraphs such that(1)Each edge of belongs to one of the for some (2)If , then and have no edges in common.
3. Prime Decomposition of Zero Divisor Graph
Theorem 1. If is any prime number, and then the graph admits a prime decomposition if and only if where is any prime less than .
Proof 1. Suppose that is any prime number, and , we have, be the nonzero zero divisor graph. The vertex set of is .
Case 1. Let us consider . If the graph is prime decomposition into copies of then there exists where is any prime number less than .
Case 2. Let us consider then their exists a remainder . If the graph is prime decomposition into copies of and (copies of copies of ) then there exists where is any prime number less than . Above-given case 1 and case 2 clearly show that the prime decomposition of into then .
Conversely, suppose that where is any prime less than .
Let us consider and . If then there exists 4 copies of . If then there exists 2 copies of and (2 copies of one copy of ).
Let us take and . If then there exists 6 copies of . If then there exists 4 copies of . If then there exists 2 copies of and (2 copies of one copy of ).
In general, take is any prime number and is prime numbers less than . Clearly, If then is a prime decomposition into copies of and (2 copies of copies of ). Hence the proof (see Figure 1).
Example 1. Let us take and the graph as the example of Theorem 1.
Theorem 2. If is any prime number, and then the graph admits a prime decomposition if and only if where is any prime less than p.
Proof 2. Suppose that is any prime number, , we have, be the nonzero zero divisor graph with isomorphic to . The vertex set of is .
Case 3. Let us take . If the prime decomposition of into copies of then there exists where is any prime numbers less than .
Case 4. Let us take . If the prime decomposition of into copies of . Clearly, the remaining edges are ( copies of 4 copies of ) where is the remainder of . Then, there exists . Clearly, shows that the above cases prime decomposition of into then .
Conversely, suppose that where is any prime numbers less than .
Let us take and . If then 12 copies of . If then 8 copies of . If then 4 copies of and (4 copies of one copy of ).
Let us take and . If then 20 copies of . If then 12 copies of and (4 copies of one copy of . If then 8 copies of . If then 4 copies of and (4 copies of 3 copies of ).
In general, take is any prime number and is prime numbers less than . Clearly, If then is a prime decomposition into copies of and (4 copies of copies of ). Hence the proof (see Figure 2).
Example 2. Let us take and the graph as the example of Theorem 2
Theorem 3. If is any prime number, and then the graph admits a prime decomposition if and only if where is any prime less than p.
Proof 3. Suppose that is any prime number, , we have, be the nonzero zero divisor graph with isomorphic to . The vertex set of is .
Case 5. Let us take the prime decomposition of into copies of then there exists where is any prime numbers less than .
Case 6. Let us take the prime decomposition of into copies of . Clearly, the remaining edges are ( copies of 6 copies of ) where is the remainder of . Then, there exists . Clearly, shows that the above-given cases prime decomposition of into then .
Conversely, suppose that where is any prime numbers less than . Let us take and . If then 30 copies of . If then 18 copies of and (6 copies of one copy of ). If then 12 copies of . If then 6 copies of and (6 copies of 3 copies of ). Let us take and . If then 36 copies of . If then 24 copies of . If then 12 copies of and (6 copies of 2 copies of ). If then 6 copies of and (6 copies of 5 copies of ). If then 6 copies of and (6 copies of one copy of ). In general, take is any prime number and is prime numbers less than . Clearly, If then is a prime decomposition into copies of and 6 copies of copies of . Hence the proof (see Figure 3).
Example 3. Let us take and the graph as the example of Theorem 3,
Theorem 4. If and are any distinct prime numbers, and then the graph admits a prime decomposition if and only if where is any prime less than p.
Proof 4. Suppose that and are any distinct prime numbers, , we have, be the nonzero zero divisor graph with isomorphic to . The vertex set of is .
Case 7. Let us take . If prime decomposition of into copies of then there exists where is any prime numbers less than .
Case 8. Let us take the prime decomposition of into copies of . Clearly, the remaining edges are ( copies of copies of ) where is the remainder of . Then, there exists . Clearly, shows that the above cases prime decomposition of into then .
Conversely, suppose that where is any prime numbers less than .
Let us take and are any distinct prime numbers and is any prime less than . Clearly, the above-given theorems show. If then is a prime decomposition into copies of and copies of copies of . 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 of , , , and are established. In the future, we will study some more properties and applications of Prime Decomposition of Zero Divisor Graph. [9].
Data Availability
The data utilized for the model development of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.