# On the planarity of a graph associated to a commutative ring and on the planarity of its complement

05 Jul 2017-The São Paulo Journal of Mathematical Sciences (Springer International Publishing)-Vol. 11, Iss: 2, pp 405-429

TL;DR: In this article, the authors consider the problem of classifying rings R such that the complement of the annihilating ideal I of R is planar, i.e., the set of all annihilating ideals of R has a planar complement.

Abstract: The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. An ideal I of R is said to be an annihilating ideal if there exists $$r\in R\backslash \{0\}$$
such that $$Ir = (0)$$
. Let $$\mathbb {A}(R)$$
denote the set of all annihilating ideals of R and let us denote $$\mathbb {A}(R)\backslash \{(0)\}$$
by $$\mathbb {A}(R)^{*}$$
. Visweswaran and Patel (Discrete Math Algorithms Appl 6:22, 2014) introduced and studied a graph, denoted by $$\Omega (R)$$
, whose vertex set is $$\mathbb {A}(R)^{*}$$
and distinct vertices I, J are joined by an edge in this graph if and only if $$I + J\in \mathbb {A}(R)$$
. In Visweswaran and Sarman (Discrete Math Algorithms Appl 8:22, 2016), we investigated some properties of the complement of $$\Omega (R)$$
. The aim of this article is to classify rings R in order that $$\Omega (R)$$
be planar. We also consider the problem of classifying rings R such that the complement of $$\Omega (R)$$
is planar.

##### Citations

More filters

••

TL;DR: Graph Theory with Applications to Engineering and Computer as mentioned in this paper is an excellent introductory treatment of graph theory and its applications that has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject.

Abstract: Graph Theory with Applications to Engineering and Computer ... This outstanding introductory treatment of graph theory and its applications has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject. The first nine chapters constitute an excellent overall introduction, requiring only some knowledge of set theory and matrix algebra.

16 citations

•

TL;DR: In this paper, the authors considered commutative with identity rings that admit at least two maximal ideals, i.e., a ring is a ring such that it admits at least 2 maximal ideals.

Abstract: The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals.

2 citations

•

01 Dec 2021TL;DR: The annihilator-inclusion ideal graph of R, denoted by ξR, is a graph whose vertex set is the of all non-zero proper ideals of R, and two distinct vertices $I$ and $J$ are adjacent if and only if either ANN(I) ⊆ J or ANN(J)⊆ I.

Abstract: Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

2 citations

••

TL;DR: This paper characterize all Artinian rings [Formula: see text] for which the genus of [Form formula]: see text is zero or one.

Abstract: Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by 𝒜R, is a graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I ...

1 citations

##### References

More filters

•

01 Jan 1969

TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:

Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

4,168 citations

•

01 Jan 1974

TL;DR: This outstanding introductory treatment of graph theory and its applications has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject.

1,161 citations

••

TL;DR: In this article, the authors present the idea of coloring of a commutative ring and show that the existence of an infinite clique implies that the clique R = co implies that there exists an infinitely many cliques.

956 citations