scholarly journals Categorial properties of compressed zero-divisor graphs of finite commutative rings

2020 ◽  
pp. 2150069
Author(s):  
Alen Đurić ◽  
Sara Jevđnić ◽  
Nik Stopar
Keyword(s):  
Principal Ideal ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Local Rings ◽  
Unital Ring ◽  
Zero Divisor Graph ◽  
Definition Of ◽  
Unital Rings ◽  
Finite Commutative Rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

On zero-divisor graphs of commutative rings without identity

2019 ◽  
Vol 19 (12) ◽  
pp. 2050226 ◽  
Author(s):  
G. Kalaimurugan ◽  
P. Vignesh ◽  
T. Tamizh Chelvam
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Free Order ◽  
Finite Commutative Rings

Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

scholarly journals Metric and upper dimension of zero divisor graphs associated to commutative rings

2020 ◽  
Vol 12 (1) ◽  
pp. 84-101 ◽  
Author(s):  
S. Pirzada ◽  
M. Aijaz
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractLet R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

scholarly journals Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Abdulaziz M. Alanazi ◽  
Mohd Nazim ◽  
Nadeem Ur Rehman
Keyword(s):  
Zero Divisor ◽  
Domination Number ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Nonzero Intersection ◽  
Toroidal Graphs ◽  
Vertex Set ◽  
Generalized Zero ◽  
Finite Commutative Rings

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z A . An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A . It is denoted by ℐ ≤ e A . The generalized zero-divisor graph denoted by Γ g A is an undirected graph with vertex set Z A ∗ (set of all nonzero zero-divisors of A ) and two distinct vertices x 1 and x 2 are adjacent if and only if ann x 1 + ann x 2 ≤ e A . In this paper, first we characterized all the finite commutative rings A for which Γ g A is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Γ g A is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Γ g A .

Finite Rings Whose Graphs Have Clique Number Less than Five

2021 ◽  
Vol 28 (03) ◽  
pp. 533-540
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Clique Number ◽  
Commutative Rings ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisor graph. We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one, two, or three. Furthermore, if [Formula: see text] (each [Formula: see text] is local for [Formula: see text]), we also give algebraic characterizations of the ring [Formula: see text] when the clique number of [Formula: see text] is four.

The maximal degree of a zero-divisor graph

2019 ◽  
pp. 2050151
Author(s):  
Husam Q. Mohammad ◽  
Nazar H. Shuker ◽  
Luma A. Khaleel
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finite Ring ◽  
Maximal Degree ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Finite Commutative Rings

The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

scholarly journals Eigenvalues of zero-divisor graphs of finite commutative rings

2020 ◽  
Author(s):  
Katja Mönius
Keyword(s):  
Prime Number ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Graph Product ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractWe investigate eigenvalues of the zero-divisor graph $$\Gamma (R)$$ Γ ( R ) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of $$\Gamma (R)$$ Γ ( R ) . The graph $$\Gamma (R)$$ Γ ( R ) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever $$xy = 0$$ x y = 0 . We provide formulas for the nullity of $$\Gamma (R)$$ Γ ( R ) , i.e., the multiplicity of the eigenvalue 0 of $$\Gamma (R)$$ Γ ( R ) . Moreover, we precisely determine the spectra of $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$ Γ ( Z p × Z p × Z p ) and $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$ Γ ( Z p × Z p × Z p × Z p ) for a prime number p. We introduce a graph product $$\times _{\Gamma }$$ × Γ with the property that $$\Gamma (R) \cong \Gamma (R_1) \times _{\Gamma } \cdots \times _{\Gamma } \Gamma (R_r)$$ Γ ( R ) ≅ Γ ( R 1 ) × Γ ⋯ × Γ Γ ( R r ) whenever $$R \cong R_1 \times \cdots \times R_r.$$ R ≅ R 1 × ⋯ × R r . With this product, we find relations between the number of vertices of the zero-divisor graph $$\Gamma (R)$$ Γ ( R ) , the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of $$\Gamma (R)$$ Γ ( R ) .

scholarly journals On planarity of compressed zero-divisor graphs associated to commutative rings

2020 ◽  
Vol 29 (2) ◽  
pp. 131-136
Author(s):  
M. IMRAN BHAT ◽  
S. PIRZADA ◽  
AHMAD M. ALGHAMDI
Keyword(s):  
Equivalence Class ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Equivalence Classes ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The equivalence class [r] of an element r ∈ R is the set of zero-divisors s such that ann(r) = ann(s), that is, [r] = {s ∈ R : ann(r) = ann(s). The compressed zero-divisor graph, denoted by Γc(R), is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring R, that is, the vertex set of Γc(R) is Re − {[0], [1]}, where Re = {[r] : r ∈ R} and two distinct equivalence classes [r] and [s] are adjacent if and only if rs = 0. In this article, we investigate the planarity of Γc(R) for some finite local rings of order p 2 , p 3 and determine the planarity of compressed zero-divisor graph of some local rings of order 32, whose zero-divisor graph is nonplanar. Further, we determine values of m and n for which Γc(Zn) and Γc(Zn[x]/(xm)) are planar.

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