Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

Let R be a commutative ring with identity. The set 𝕀(R) of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of 𝕀(R) is called the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). We write 𝒢 for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in 𝒢 that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that 𝔸𝔾(R) ∈ 𝒢 is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with 𝔸𝔾(R) ∈ 𝒢. Also, a connection between finite rings and their corresponding graphs is realized.     

Zero-Divisor Graph with Respect to an Ideal

ABSTRACT

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ _I (R), is the graph whose vertices are the set {x ? R\I | xy ? I for some y ? R\I} with distinct vertices x and y adjacent if and only if xy ? I. In the case I = 0, Γ₀(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ _I (R) ? Γ _J (S) and Γ(R/I) ? Γ(S/J). We also discuss when Γ _I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ _I (R).     

On a graph of ideals

We associate a graph Γ₊(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I+J=R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ₊(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ₊(R).     

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)?{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ _I (R), with vertices {x ∈ R?I | xy ∈ I for some y ∈ R?I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ _I (R). We also determine when Γ _I (R) is either a complete graph or a complete bipartite graph and investigate when Γ _I (R) ? Γ(S) for some commutative ring S.     

The Inclusion Ideal Graph of Rings

Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ? J or J ? I. In this paper, we show that In(R) is not connected if and only if R ? M ₂(D) or D ₁ × D ₂, for some division rings, D, D ₁ and D ₂. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.     

Some Properties of a Cayley Graph of a Commutative Ring

Let R be a commutative ring with unity and R ⁺, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ?𝔸𝕐(R) and G _R , the Cayley graph Cay(R ⁺, Z*(R)) and the unitary Cayley graph Cay(R ⁺, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied G _R . In this article, we study ?𝔸𝕐(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ?𝔸𝕐(R). We also characterize all rings R whose ?𝔸𝕐(R) is planar. Moreover, we determine all finite rings R whose ?𝔸𝕐(R) is strongly regular. We prove that ?𝔸𝕐(R) is strongly regular if and only if it is edge transitive. As a consequence, we characterize all finite rings R for which G _R is a strongly regular graph.     

Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by Γ_I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ? V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of Γ_I (R) and the comaximal graph CG₂(R) \ J (R) (or CGJ (R) for short) where CG₂(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).     

Algebraic Properties of the Path Ideal of a Tree

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I _t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I _t (Γ) when Γ is a tree. We first show that I _t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I _t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I _t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.     

When a line graph associated to annihilating-ideal graph of a lattice is planar or projective

Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if I ∧ J = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.     

On the Annihilator Graph of a Commutative Ring

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann _R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R)?{0}, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.     

On the Comparison of Rees's Valuations Over a Quasi-Unmixed Local Ring

In this paper, we will show that if (R, 𝔪) is a quasi-unmixed local ring, I an 𝔪-primary ideal of R and ?𝒱(I) is the set of Rees valuations of I, then the number of minimal prime ideals in the 𝔪-adic completion of R equals exactly the number of equivalence classes on the set ?𝒱(I) under the equivalence relation ～defined by: ν₁ ～ ν₂ if there exist a constant c ≥ 1 such that for all x ∈ R, ν₁(x) ≤ cν₂(x) and ν₂(x) ≤ cν₁(x).     

Coefficient Ideal of Ideals Generated by Monomials

In a commutative Noetherian ring R, the coefficient ideal of I relative to J is the largest ideal 𝔟 for which I𝔟 =J𝔟 when I is integral over J. In this article, we will give a simple algorithm to compute 𝔞(I, J) when I, J are ideals in a polynomial ring R = k[X ₁,…, X _d ] generated by monomials and J is a parameter ideal. We use the concept of socle sequence. Also we will show that the reduction number r _J (I) is also computed by our algorithm.     

On the rank spread of graphs

For a simple graph G?=?(𝒱, ?) with vertex-set 𝒱?=?{1,?…?,?n}, let 𝒮(G) be the set of all real symmetric n-by-n matrices whose graph is G. We present terminology linking established as well as new results related to the minimum rank problem, with spectral properties in graph theory. The minimum rank mr(G) of G is the smallest possible rank over all matrices in 𝒮(G). The rank spread r _v (G) of G at a vertex v, defined as mr(G)???mr(G???v), can take values ??∈?{0,?1,?2}. In general, distinct vertices in a graph may assume any of the three values. For ??=?0 or 1, there exist graphs with uniform r _v (G) (equal to the same integer at each vertex v). We show that only for ??=?0, will a single matrix A in 𝒮(G) determine when a graph has uniform rank spread. Moreover, a graph G, with vertices of rank spread zero or one only, is a λ-core graph for a λ-optimal matrix A in 𝒮(G). We also develop sufficient conditions for a vertex of rank spread zero or two and a necessary condition for a vertex of rank spread two.     

On the Dot Product Graph of a Commutative Ring

Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R?{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)?{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD(R) is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to ?₂ × ?₂ × ?₂.     

On the Zero-Divisor Graph of a Ring

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ? zR for all z ∈ Z(R)\Nil(R).     

On the Cubicity of Interval Graphs

A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where R _i (for 1 ≤ i ≤ k) is an interval of the form [a _i , a _i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to .     

Multiplicities and Reduction Numbers

Let (Rmbe a Cohen–Macaulay local ring and let I be an ideal. There are at least five algebras built on I whose multiplicity data affect the reduction number r(I) of the ideal. We introduce techniques from the Rees algebra theory of modules to produce estimates for r(I), for classes of ideals of dimension one and two. Previous cases of such estimates were derived for ideals of dimension zero.     

On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.     

The Ideal Completion of a Noetherian Local Domain

The ideal topology on a integral domain R is the linear topology which has as a fundamental system of neighborhoods of 0 the nonzero ideals of R. We investigate the properties of the ideal topology on a Noetherian local domain (R, 𝔪), and we establish connections between the 𝔪-adic completion and the ideal completion. We give conditions under which the completion in the ideal topology is Noetherian, and we show that, unlike the 𝔪-adic completion, the completion in the ideal topology is not always Noetherian.