CHARACTERIZATION OF RINGS WITH PLANAR TOROIDAL OR PROJECTIVE PLANAR PRIME IDEAL SUM GRAPHS PRAVEEN MATHIL BARKHA BALODA JITENDER KUMAR A. SOMASUNDARAM

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CHARACTERIZATION OF RINGS WITH PLANAR, TOROIDAL OR PROJECTIVE

PLANAR PRIME IDEAL SUM GRAPHS

PRAVEEN MATHIL†, BARKHA BALODA†, JITENDER KUMAR†,∗, A. SOMASUNDARAM§

Abstract. Let Rbe a commutative ring with unity. The prime ideal sum graph PIS(R) of the ring Ris the simple

undirected graph whose vertex set is the set of all nonzero proper ideals of Rand two distinct vertices Iand Jare

adjacent if and only if I+Jis 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 Rsuch that PIS(R) 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.

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, [2, 4, 6, 7, 8, 14, 15, 17, 27]). 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 [3],

intersection graphs of ideals [18], ideal-relation graph [26], co-maximal ideal graph [44], prime ideal sum graph [34]

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 [5, 12, 16, 38, 42, 43].

Genus and crosscap of the total graph of the ring were investigated in [11, 23, 28, 36]. Asir et al. [10] determined

all isomorphic classes of commutative rings whose ideal based total graph has genus at most two. Pucanovi´c et al.

[30] classiﬁed the planar and toroidal graphs that are intersection ideal graphs of Artinian commutative rings. In

[31], all the graphs of genus two that are intersection graphs of ideals of some commutative rings are characterized.

Ramanathan [32] 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 [1, 9, 19, 20, 21, 22, 24, 25, 33, 35, 37].

Recently, Saha et al. [34] introduced and studied the prime ideal sum graph of a commutative ring. The prime

ideal sum graph PIS(R) of the ring Ris the simple undirected graph whose vertex set is the set of all nonzero proper

ideals of Rand two distinct vertices Iand Jare adjacent if and only if I+Jis a prime ideal of R. Authors of

[34] studied the interplay between graph-theoretic properties of PIS(R) and algebraic properties of ring R. In this

connection, they investigated the clique number, the chromatic number and the domination number of prime ideal

2020 Mathematics Subject Classiﬁcation. 05C25.

Key words and phrases. Non-local ring, ideals, genus and crosscap of a graph, prime ideal.

* Corresponding author

†Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani-333031 (Rajasthan), India

§Department of General Sciences, Birla Institute of Technology and Science, Pilani, Dubai Campus, Dubai, UAE

Email address: maithilpraveen@gmail.com, barkha0026@gmail.com, jitenderarora09@gmail.com, asomasundaram@dubai.bits-

pilani.ac.in.

arXiv:2210.15335v2 [math.CO] 18 Apr 2024

2 PRAVEEN MATHIL, BARKHA BALODA, JITENDER KUMAR, A. SOMASUNDARAM

sum graph PIS(R). The purpose of this article is to investigate the prime ideal sum graph PIS(R) to a greater extent.

In this connection, we discuss the question of embedding of PIS(R) on various surfaces without edge crossing. This

paper aims to characterize all commutative non-local Artinian rings for which PIS(R) has crosscap at most two. We

also characterise 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 PIS(R) 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 Rfor which PIS(R) has genus one. Also, we determine all the non-local

commutative Artinian rings for which PIS(R) has crosscap at most two.

2. preliminaries

Agraph Γ is a pair (V(Γ), E(Γ)), where V(Γ) and E(Γ) are the set of vertices and edges of Γ, respectively. Two

distinct vertices u1and u2are adjacent , denoted by u1∼u2(or (u1, u2)), if there is an edge between u1and u2.

Otherwise, we write as u1≁u2. If X⊆V(Γ) then the subgraph Γ(X) induced by Xis the graph with vertex set X

and two vertices of Γ(X) are adjacent if and only if they are adjacent in Γ. For other basic graph theoretic deﬁnitions

and concepts, we refer the reader to [39, 40]. 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. [39] A graph Γis outerplanar if and only if it does not contain a subdivision of K4or K2,3.

Theorem 2.2. [39] A graph Γis planar if and only if it does not contain a subdivision of K5or K3,3.

A compact connected topological space such that each point has a neighbourhood homeomorphic to an open disc

is called a surface. For a non-negative integer g, let Sgbe the orientable surface with ghandles. The genus g(Γ) of a

graph Γ is the minimum integer gsuch that the graph can be embedded in Sg, i.e. the graph Γ can be drawn into a

surface Sgwith 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. [40, Ringel and Youngs] Let m, n be positive integers.

(i) If n≥3, then g(Kn) = l(n−3)(n−4)

12 m.

(ii) If m, n ≥2, then g(Km,n) = l(m−2)(n−2)

4m.

Lemma 2.4. [40, Theorem 5.14] Let Γbe a connected graph with a 2-cell embedding in Sg. Then v−e+f= 2 −2g,

where v, e and fare the number of vertices, edges and faces embedded in Sg, respectively and gis the genus of the

graph Γ.

Lemma 2.5. [41] The genus of a connected graph Γis the sum of the genera of its blocks.

Let Nkdenote the non-orientable surface formed by the connected sum of kprojective planes, that is, Nkis a

non-orientable surface with kcrosscap. The crosscap of a graph Γ, denoted by cr(Γ), is the minimum non-negative

integer ksuch that Γ can be embedded in Nk. For instance, a graph Γ is planar if cr(Γ) = 0 and the Γ is projective

if cr(Γ) = 1. The following results are useful to obtain the crosscap of a graph.

CHARACTERIZATION OF RINGS WITH PLANAR, TOROIDAL OR PROJECTIVE PLANAR PRIME IDEAL SUM GRAPHS 3

Proposition 2.6. [29, Ringel and Youngs] Let m, n be positive integers. Then

(i) cr(Kn) = 





l(n−3)(n−4)

6mif n≥3

3if n= 7

(ii) cr(Km,n) = l(m−2)(n−2)

2mif m, n ≥2

Lemma 2.7. [29, Lemma 3.1.4] Let ϕ: Γ →Nkbe a 2-cell embedding of a connected graph Γto the non-orientable

surface Nk. Then v−e+f= 2 −k, where v, e and fare the number of vertices, edges and faces of ϕ(Γ) respectively,

and kis the crosscap of Nk.

Deﬁnition 2.8. [41] A graph Γ is orientably simple if µ(Γ) ̸= 2 −cr(Γ), where µ(Γ) = max{2−2g(Γ),2−cr(Γ)}.

Lemma 2.9. [41] Let Γbe a graph with blocks Γ1,Γ2,· · · ,Γk. Then

cr(Γ) =











1−k+

i=1

cr(Γi),if Γis orientably simple

2k−

i=1

µ(Γi),otherwise.

We use the following remark frequently in this paper.

Remark 2.10.For a simple graph Γ, we have 2e≥3f.

A ring Ris called local if it has a unique maximal ideal Mand it is abbreviated by (R, M). For an ideal Iof R,

the smallest positive integer nsuch that In= 0 is called the nilpotency index η(I) of the ideal I. Let Rbe a non-local

commutative Artinian ring. By the structural theorem (see [13]), Ris uniquely (up to isomorphism) a ﬁnite direct

product of local rings Rithat is R∼

=R1×R2× · · · × Rn, where n≥2. Note that for any commutative Artinian ring,

every prime ideal is a maximal ideal (see [13, 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 Max(R) denotes the set of all maximal ideals of R. The set of non-zero proper

ideals of Ris denoted by I∗(R). Throughout the paper, Fidenotes a ﬁeld. For other basic deﬁnitions of ring theory,

we refer the reader to [13].

Remark 2.11.Let R∼

=R1×R2× · · · × Rn(n≥2) be a non-local commutative ring, where each Riis a local ring

with maximal ideal Mi. Then Max(R) = {J1, J2, . . . , Jn}, where Ji=R1×R2× · · · × Ri−1× Mi×Ri+1 × · · · × Rn.

3. Embedding of PIS(R)on surfaces

In this section, we study the embedding of the prime ideal sum graph PIS(R) on a surface without edge crossing.

We begin with the investigation of an embedding of PIS(R) on a plane.

3.1. Planarity of PIS(R).In this subsection, we classify all the non-local commutative Artinian rings with unity

for which the graph PIS(R) is planar and outerplanar, respectively.

Theorem 3.1. Let R∼

=R1×R2× · · · × Rn(n≥2) be a non-local commutative ring, where each Riis a local

ring with maximal ideal Miand let F1,F2,F3be ﬁelds. Then the graph PIS(R)is planar if and only if one of the

following holds:

(i) R∼

=F1×F2×F3.

(ii) R∼

=F1×F2.

(iii) R∼

=R1×R2such that both R1and R2are principal ideal rings with η(M1) = η(M2) = 2.

(iv) R∼

=F1×R2, where the local ring R2is principal ideal ring.

4 PRAVEEN MATHIL, BARKHA BALODA, JITENDER KUMAR, A. SOMASUNDARAM

Proof. First suppose that the graph PIS(R) is planar and R∼

=R1×R2× · · · × Rn, where n≥5. Then the set

X={M1×R2× · · · × Rn,M1× ⟨0⟩ × R3× · · · × Rn,M1×R2× ⟨0⟩ × R4× · · · × Rn,M1×R2×R3× ⟨0⟩ ×

R5× · · · × Rn,M1×R2×R3×R4× ⟨0⟩ × R6× · · · × Rn}induces a subgraph isomorphic to K5, a contradiction.

Therefore, n∈ {2,3,4}. If R∼

=R1×R2×R3×R4, then by Figure 1 and Theorem 2.2, the graph PIS(R) is not

planar, a contradiction. Consequently, n∈ {2,3}.

R1× M2× M3× M4

M1×R2× M3× M4

M1× M2×R3× M4

M1×R2×R3× M4

R1×R2×R3× M4

R1× M2×R3× M4

R1× M2×R3×R4

M1× M2×R3×R4

M1×R2×R3×R4

M1×R2× M3×R4

Figure 1. A subgraph of PIS(R1×R2×R3×R4) homeomorphic to K3,3

Now assume that R∼

=R1×R2×R3such that one of Ri(1 ≤i≤3) is not a ﬁeld. Without loss of generality, we

assume that R1is not a ﬁeld. By Figure 2, note that PIS(R) contains a subgraph homeomorphic to K5, which is

not possible. Thus, R∼

=F1×F2×F3. We may now suppose that R∼

=R1×R2. First note that both R1and R2

M1× ⟨0⟩×⟨0⟩

M1× ⟨0⟩ × F3

M1×F2× ⟨0⟩

M1×F2×F3

⟨0⟩ × F2×F3

R1×F2× ⟨0⟩

R1× ⟨0⟩ × F3

Figure 2. A subgraph of PIS(R1×F2×F3) homeomorphic to K5

are principal rings. On contrary, if one of them is not principal say R1. Then R1has atleast two nontrivial ideals

J1and J2diﬀrent from M1such that J1+J2=M1. By Figure 3, note that PIS(R1×R2) contains a subgraph

homeomorphic to K3,3, which is not possible. Consequently, R∼

=R1×R2such that both the rings R1and R2are

principal. If at least one of Ri(1 ≤i≤2) is a ﬁeld, then Ris isomorphic to F1×F2or F1×R2, where R2is a

principal ideal ring. Now let R∼

=R1×R2such that both R1and R2are principal ideal rings which are not ﬁelds.

If η(M1)≥3 and η(M2)≥2, then there exists a non-zero proper ideal M2

1(̸=M1) of R1. By Figure 4, the graph

PIS(R1×R2) contains a subgraph homeomorphic to K3,3, a contradiction. Consequently, for R∼

=R1×R2, where

both R1and R2are not ﬁelds, we have η(M1) = η(M2) = 2.

The converse follows from Figures 5, 6, 7 and 8. □

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CHARACTERIZATIONOFRINGSWITHPLANAR,TOROIDALORPROJECTIVEPLANARPRIMEIDEALSUMGRAPHSPRAVEENMATHIL†,BARKHABALODA†,JITENDERKUMAR†,∗,A.SOMASUNDARAM§Abstract.LetRbeacommutativeringwithunity.TheprimeidealsumgraphPIS(R)oftheringRisthesimpleundirectedgraphwhosevertexsetisthesetofallnonzeroproperidealsofRandtwod...

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CHARACTERIZATION OF RINGS WITH PLANAR TOROIDAL OR PROJECTIVE PLANAR PRIME IDEAL SUM GRAPHS PRAVEEN MATHIL BARKHA BALODA JITENDER KUMAR A. SOMASUNDARAM

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