Abstract
Let R be a commutative ring with unity. The prime ideal sum graph of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings R such that is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
1 Introduction
The investigation of algebraic structures through graph-theoretic properties of the associated graphs has become a fascinating research subject over the past few decades. Numerous graphs attached with ring structures have been studied in the literature (see, [Citation2, Citation4, Citation6–8, Citation14, Citation15, Citation17, Citation27]). Because of the crucial role of ideals in the theory of rings, many authors have explored the graphs attached to the ideals of rings also, for example, inclusion ideal graph [Citation3], intersection graphs of ideals [Citation18], ideal-relation graph [Citation26], co-maximal ideal graph [Citation44], prime ideal sum graph [Citation34] etc. Topological graph theory is principally related to the embedding of a graph on a surface without edge crossing. Its applications lie in electronic printing circuits where the purpose is to embed a circuit, that is, the graph on a circuit board (the surface) without two connections crossing each other, resulting in a short circuit. To determine the genus and crosscap of a graph is a fundamental but highly complex problem. Indeed, it is NP-complete. Many authors have investigated the problem of finding the genus of zero divisor graphs of rings in [Citation5, Citation12, Citation16, Citation38, Citation42, Citation43]. Genus and crosscap of the total graph of the ring were investigated in [Citation11, Citation23, Citation28, Citation36]. Asir et al. [Citation10] determined all isomorphic classes of commutative rings whose ideal based total graph has genus at most two. Pucanović et al. [Citation30] classified the planar and toroidal graphs that are intersection ideal graphs of Artinian commutative rings. In [Citation31], all the graphs of genus two that are intersection graphs of ideals of some commutative rings are characterized. Ramanathan [Citation32] determined all Artinian commutative rings whose intersection ideal graphs have crosscap at most two. Work related to the embedding of graphs associated with other algebraic structures on a surface can be found in [Citation1, Citation9, Citation19–22, Citation24, Citation25, Citation33, Citation35, Citation37].
Recently, Saha et al. [Citation34] introduced and studied the prime ideal sum graph of a commutative ring. The prime ideal sum graph of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. Authors of [Citation34] studied the interplay between graph-theoretic properties of and algebraic properties of ring R. In this connection, they investigated the clique number, the chromatic number and the domination number of prime ideal sum graph . The purpose of this article is to investigate the prime ideal sum graph to a greater extent. In this connection, we discuss the question of embedding of on various surfaces without edge crossing. This paper aims to characterize all commutative non-local Artinian rings for which has crosscap at most two. We also characterize all the non-local commutative Artinian rings whose prime ideal sum graph is toroidal. Moreover, we classify all the non-local commutative Artinian rings for which is planar and outerplanar, respectively. The paper is arranged as follows. Section 2 comprises basic definitions and necessary results. In Section 3, we classify all the non-local commutative Artinian rings R for which has genus one. Also, we determine all the non-local commutative Artinian rings for which has crosscap at most two.
2 Preliminaries
A graph Γ is a pair , where and are the set of vertices and edges of Γ, respectively. Two distinct vertices u1 and u2 are , denoted by (or ), if there is an edge between u1 and u2. Otherwise, we write as . If then the subgraph induced by X is the graph with vertex set X and two vertices of are adjacent if and only if they are adjacent in Γ. For other basic graph theoretic definitions and concepts, we refer the reader to [Citation39, Citation40]. A graph Γ is outerplanar if it can be embedded in the plane such that all vertices lie on the outer face of Γ. In a graph Γ, the subdivision of an edge (u, v) is the deletion of (u, v) from Γ and the addition of two edges (u, w) and (w, v) along with a new vertex w. A graph obtained from Γ by a sequence of edge subdivision is called a subdivision of Γ. Two graphs are said to be homeomorphic if both can be obtained from the same graph by subdivisions of edges. A graph Γ is planar if it can be drawn on a plane without edge crossing. It is well known that every outerplanar graph is a planar graph. The following results will be useful for later use.
Theorem 2.1.
[Citation39] A graph Γ is outerplanar if and only if it does not contain a subdivision of K4 or .
Theorem 2.2.
[Citation39] A graph Γ is planar if and only if it does not contain a subdivision of K5 or .
A compact connected topological space such that each point has a neighborhood homeomorphic to an open disc is called a surface. For a nonnegative integer g, let be the orientable surface with g handles. The genus of a graph Γ is the minimum integer g such that the graph can be embedded in , i.e. the graph Γ can be drawn into a surface with no edge crossing. Note that the graphs having genus 0 are planar and the graphs having genus one are toroidal. The following results are useful in the sequel.
Proposition 2.3.
[Citation40, Ringel and Youngs] Let m, n be positive integers.
If , then .
If , then .
Lemma 2.4.
[Citation40, Theorem 5.14] Let Γ be a connected graph with a 2-cell embedding in . Then , where v, e and f are the number of vertices, edges and faces embedded in , respectively and g is the genus of the graph Γ.
Lemma 2.5.
[Citation41] The genus of a connected graph Γ is the sum of the genera of its blocks.
Let denote the non-orientable surface formed by the connected sum of k projective planes, that is, is a non-orientable surface with k crosscap. The crosscap of a graph Γ, denoted by , is the minimum nonnegative integer k such that Γ can be embedded in . For instance, a graph Γ is planar if and the Γ is projective if . The following results are useful to obtain the crosscap of a graph.
Proposition 2.6.
[Citation29, Ringel and Youngs] Let m, n be positive integers. Then
Lemma 2.7.
[Citation29, Lemma 3.1.4] Let be a 2-cell embedding of a connected graph Γ to the non-orientable surface . Then , where v, e and f are the number of vertices, edges and faces of , respectively, and k is the crosscap of .
Definition 2.8.
[Citation41] A graph Γ is orientably simple if , where .
Lemma 2.9.
[Citation41] Let Γ be a graph with blocks . Then
We use the following remark frequently in this paper.
Remark 2.10.
For a simple graph Γ, we have .
A ring R is called local if it has a unique maximal ideal and it is abbreviated by . For an ideal I of R, the smallest positive integer n such that is called the nilpotency index of the ideal I. Let R be a non-local commutative Artinian ring. By the structural theorem (see [Citation13]), R is uniquely (up to isomorphism) a finite direct product of local rings Ri that is , where . Note that for any commutative Artinian ring, every prime ideal is a maximal ideal (see [Citation13, Proposition 8.1]). Hence, we use the maximal ideal for the adjacency between the two vertices of the prime ideal sum graph. The following remark specifies maximal ideals in a non-local Artinian commutative ring, where denotes the set of all maximal ideals of R. The set of nonzero proper ideals of R is denoted by . Throughout the paper, Fi denotes a field. For other basic definitions of ring theory, we refer the reader to [Citation13].
Remark 2.11.
Let be a non-local commutative ring, where each Ri is a local ring with maximal ideal . Then , where .
3 Embedding of on surfaces
In this section, we study the embedding of the prime ideal sum graph on a surface without edge crossing. We begin with the investigation of an embedding of on a plane.
3.1 Planarity of
In this subsection, we classify all the non-local commutative Artinian rings with unity for which the graph is planar and outerplanar, respectively.
Theorem 3.1.
Let be a non-local commutative ring, where each Ri is a local ring with maximal ideal and let F1, F2, F3 be fields. Then the graph is planar if and only if one of the following holds:
.
.
such that both R1 and R2 are principal ideal rings with .
, where the local ring R2 is principal ideal ring.
Proof.
First suppose that the graph is planar and , where . Then the set induces a subgraph isomorphic to K5, a contradiction. Therefore, . If , then by and Theorem 2.2, the graph is not planar, a contradiction. Consequently, .
Now assume that such that one of is not a field. Without loss of generality, we assume that R1 is not a field. By , note that contains a subgraph homeomorphic to K5, which is not possible. Thus, .
We may now suppose that . First note that both R1 and R2 are principal rings. On contrary, if one of them is not principal say R1. Then R1 has atleast two nontrivial ideals J1 and J2 different from such that . By , note that contains a subgraph homeomorphic to , which is not possible.
Consequently, such that both the rings R1 and R2 are principal. If at least one of Ri is a field, then R is isomorphic to or , where R2 is a principal ideal ring. Now let such that both R1 and R2 are principal ideal rings which are not fields. If and , then there exists a nonzero proper ideal of R1. By , the graph contains a subgraph homeomorphic to , a contradiction. Consequently, for , where both R1 and R2 are not fields, we have .
The converse follows from . □
Theorem 3.2.
Let be a non-local commutative ring, where each Ri is a local ring with maximal ideal . Then the graph is outerplanar if and only if R is isomorphic to one of the following rings: , where F1, F2, F3 are fields and R2 is a principal ideal ring with .
Proof.
Suppose that is an outerplanar graph. Since every outerplanar graph is planar, we get R is isomorphic to the following rings: , where is any principal ideal ring and R1, R2 are principal ideal rings with .
Let , where R1 and R2 are principal rings. First suppose that both R1 and R2 are not fields together with and . Consider . Notice that the subgraph induced by the set of vertices of contains as a subgraph, a contradiction. Without loss of generality, we may now assume that R1 is a field. Further, let R2 is a principal ideal ring with . By , the graph contains a subgraph homeomorphic to , a contradiction. Therefore, either or , where R2 is a principal ideal ring with .
The converse follows from , , and . □
3.2 Genus of
In this subsection, we classify all the non-local commutative Artinian rings with unity R such that has toroidal embedding. By , we mean . We begin with the following lemma.
Lemma 3.3.
Let , where each Fi is a field. Then .
Proof.
Let . Consider the subset of vertices of . Note that the subgraph induced by X contains a subgraph homeomorphic to (see ). By Proposition 2.3, we have . □
Theorem 3.4.
Let be a non-local commutative ring, where each Ri is a local ring with maximal ideal . Then if and only if , where R1 is a principal ideal ring with and F2, F3 are fields.
Proof.
First suppose that . By Lemma 3.3, we have . Let . Assume that each Ri () is a field. Notice that v = 14 and e = 48. By Lemma 2.4, we obtain f = 34. It follows that , a contradiction. Consequently, . If each Ri is a field, then by , is a planar graph, which is not possible. Next, assume that R1 and R2 are not fields. Then by , the graph contains a subgraph homeomorphic to , a contradiction.
It follows that R1 is not a field but R2 and R3 are fields. Now assume that either R1 is not a principal ideal ring or R1 is a principal ideal ring with . Then . Let K be a nonzero proper ideal of R1. Then by and Lemma 2.5, we get , a contradiction.
Therefore, , where R1 is a principal ideal ring with . The converse follows from . □
Theorem 3.5.
Let , where R1 and R2 are local rings with maximal ideals and , respectively. Then if and only if R1 and R2 are principal ideal rings with and .
Proof.
Suppose that . Assume that one of Ri is not a principal ideal ring. Without loss of generality, let R1 is not a principal ideal ring. Then , where . Let r = 2 i.e., for some . Now let . Consider the vertices and . Note that the subgraph induced by these vertices contains a subgraph homeomorphic to (see ).
It follows that , a contradiction. Next let, R2 be a field. Let . Then consider the ideals and . Then the subgraph induced by these vertices contains a subgraph homeomorphic to , where and are partition sets and K1 join the vertices J5 and . By Proposition 2.3, we get a contradiction. Consequently, and so . Next, we find all the nontrivial ideals of the ring R1. Let , for some , be the principal ideal of R1 except and . Clearly both . If , then for some . Suppose that , then we have . Therefore, principal ideals of R1 are of the form where . Note that is the only ideal which is not principal. Therefore, all the nontrivial ideals of R1 are , where . If , then there exists and . Now, we show that . Clearly, . Since , we have is a unit of R1 and so . It follows that . Similarly, . Consequently, . Let . It follows that has excatly 5 vertices and . Thus, are the vertices of .
Now, consider . Note that the subgraph is homeomorphic to . By Proposition 2.3, we have and by Lemma 2.4, we get three faces in any embedding of in . Consequently, any embedding of Ω has at least one face of length 6. If we insert the adjacent vertices J2, J9 and the respective edges incident to J2 and J9 to the embedding of Ω, then (J2, J9) must be embedded in a face of length 6. To embed the graph in , we need to add the edges and remaining vertices to the embedding of . Observe that this is not possible without edge crossings. It follows that , a contradiction. If , then by the similar argument, we get a contradiction. Similarly, if , we get a contradiction. Therefore, the local rings R1 and R2 are principal.
Let , where both R1 and R2 are principal ideal rings. If , then by Theorem 3.1, we get a contradiction. Next, let . Then contains a subgraph with two blocks of (see ). By Proposition 2.3 and Lemma 2.5, we have , a contradiction.
Now assume that R1 is a principal ideal ring and R2 is a field. By Theorem 3.1, we get is a planar graph, a contradiction. If and , then by and Lemma 2.5, we obtain , a contradiction.
Therefore, , where R1 and R2 are principal ideal rings with and . The converse follows from . □
3.3 Crosscap of
In this subsection, we classify all the non-local commutative Artinian rings R with unity such that the crosscap of is at most two. The following theorem asserts that for a non-local commutative ring R its prime ideal sum graph cannot be of crosscap one.
Theorem 3.6.
For any non-local commutative Artinian ring R with unity, the prime ideal sum graph can not be projective planar.
Proof.
Let (), where each Ri is local ring with maximal ideal . Suppose that . Let . By the proof of Lemma 3.3, we get a subgraph of which is homeomorphic to . By Proposition 2.6, we get . Now suppose that . Then we get v = 14 and e = 48. Lemma 2.7 yields f = 35. It follows that , a contradiction. Thus, . We may now suppose that . If each Ri is a field, then by Theorem 3.1, we get a contradiction. Next, assume that both R1 and R2 are not fields but R3 is a field. By , the graph contains a subgraph homeomorphic to , a contradiction. Now suppose that R1 is not a field but both R2 and R3 are fields. If , then by and Lemma 2.9, we get , a contradiction. Therefore, , where . Now consider the set and the subgraph . Then G is homeomorphic to . By Lemma 2.6, we have cr(G) = 1. Moreover, by Lemma 2.7, we get four faces in any embedding of G in . Consequently, any embedding of G in has three faces of length 4 and one face of length 6. To embed from G in , first we insert vertex in an embedding of G in . Both the adjacent vertices and must be inserted in the same face . Note that is adjacent to . Also, . Moreover, . Consequently, the face should be of length 6. After inserting in , we need to add the edges , which is not possible without edge crossing (see ), a contradiction. Thus, .
Now let . Assume that one of Ri is not a principal ideal ring. Without loss of generality, let R1 is not a principal ideal ring. In the similar lines of the proof of Theorem 3.5, we get . Therefore, , where the local rings R1 and R2 are principal. If both R1 and R2 are fields, then the graph is planar, a contradiction. If , then by Theorem 3.1, the graph is planar, again a contradiction. Next, let . Then contains a subgraph which has two blocks (see ). By Proposition 2.6 and Lemma 2.9, we get , a contradiction. Now assume that R1 is a principal ideal ring and R2 is a field. By Theorem 3.1, the graph is planar, a contradiction. Next, let and . Then by and Lemma 2.9, we obtain , a contradiction. Now, let , where R1, R2 are principal ideal rings with and . Then consider the set and the subgraph . Then the subgraph is homeomorphic to . It follows that any embedding of in has 4 faces. Now to embed in through , first we insert the vertex . Since is adjacent to and , we must have and in the same face . Note that is adjacent to and . Also is adjacent to . Therefore, the face contains the vertices and . Now after inserting the edges and , the insertion of the edges without edge crossing is not possible. Hence, . □
Theorem 3.7.
Let R be a non-local commutative ring such that , where each Ri is a local ring with maximal ideal . Then if and only if
, where R1 is a principal ideal ring with and F2, F3 are fields.
, where both R1 and R2 are principal ideal rings with and .
Proof.
Suppose that . If , then by the proof of the Lemma 3.3, we get a subgraph of homeomorphic to . By the Proposition 2.6, we get . Now let . Then we get v = 14 and e = 48. By Lemma 2.7, we have f = 34. It implies that , a contradiction. Thus, . Let . Suppose that both R1 and R2 are not fields but R3 is a field. Then by , contains a subgraph homeomorphic to , a contradiction. Next, let such that . Then by , note that contains a subgraph G, which has two blocks. Then . To embed in from G, first we insert the edges and . After adding the edges e1 and e2, note that we have two triangles: (i) , and (ii) . Also is adjacent to and . Since the vertex is adjacent to and , we cannot insert the edge e3 without edge crossing. It implies that . Note that if R1 is not a principal ideal ring, then . Therefore, such that R1 is a principal ideal ring with .
Next, let . First, assume that one of Ri is not a principal ideal ring. Without loss of generality, let R1 is not a principal ideal ring. Then , where . Let r = 2 i.e., for some . Now let . By the proof of Theorem 3.5, contains a subgraph homeomorphic to , a contradiction. Let and . If , then by the proof of Theorem 3.5, contains a subgraph homeomorphic to , which is not possible. Therefore, and such that . Let be the vertices of . Consider the set . Then the subgraph induced the set is isomorphic to K5. Thus, . Observe that the subgraph is isomorphic to a subgraph of K6 and all the faces of K6 in are triangles. Thus, . Note that has 6 vertices and 13 edges. By Lemma 2.7, it follows that f = 8 in any embedding of in . Therefore, has either seven faces of length 3 and one face of length 5, or six faces of length 3 and two faces of length 4. Notice that the 5-length face of contains the vertices , and J11. Now first we insert J2 and J4 in an embedding of in . Since , they must be inserted in the same face. Note that J2 is adjacent to , and . Also, J11 is adjacent to , and J10. Moreover, , and . It follows that both J2 and J4 must be inserted in a face of length 5 containing the vertices which is not possible. Consequently, . Next, consider the set . Since , we have . For the subgraph , note that v = 9 and e = 21. It implies that f = 12 in any embedding of in . Thus, have seven faces of length 3, two faces of length 4 and one face of length 5 in any embedding of in . Now to embed in , we insert J3 in a face F of an embedding of . Then F must contain J4, J7, J8 and J10, but J4 is not adjacent to J7, J8 and J10. Thus, F must contain at least 6 vertices, which is not possible. It follows that .
Consequently, , where R1 and R2 are principal local rings. If both R1 and R2 have at most one nontrivial ideal i.e., , then by Theorem 3.1, we get a contradiction. Next, let . Note that contains a subgraph G1 as two blocks of (see ). By Lemma 2.9, . To embed in from G1, first we insert the edges and . After inserting the edges and , note that we have two triangles: (i) , and (ii) . Also, is adjacent to . Moreover, . Since , we cannot insert the edge without edge crossing. Now, let and . Then contains a subgraph which has two blocks of (see ). Then by Lemma 2.9, . To embed in through , first we insert the edges and . Note that . Also, . Then after inserting the edges , and , we can not insert the edge without edge crossing in an embedding of in . Hence, , where R1, R2 are principal ideal rings with and .
Converse follows from and . □
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We would like to thank the referees for their valuable suggestions which helped us to improve the presentation of the paper.
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