Narrow Results

One of the central themes in the research of function rings over topological spaces is build new rings, build new spaces. In this paper, we continue the work of Knox on the ring of density continuous functions, denoted by $C(X,{ℝ}_d)$. We infer that $C(X,{ℝ}_d)$ is self-injective (${\aleph }_0$-self-injective) if the family of zero-sets, denoted by $Z_d[X]$ is closed under arbitrary (countable) intersections. We provide a counterexample of the converse statement on self-injectivity, and in that process, we answer a question raised in [H. Azadi, M. Henriksen and E. Momtahan, Some properties of algebra of real valued measurable functions, Acta Math. Hunger124(1–2) (2009) 15–23]. We have established a condition on $X$ for which $C(X,{ℝ}_d)$ is Artinian and Noetherian. Unlike $C(X)$, the zero-divisor graph of $C(X,{ℝ}_d)$ is seen to be neither triangulated nor hyper-triangulated. We prove that space $X$ is density compact if and only if it is density pseudocompact and density real compact.

For a lattice $L$, we associate a graph called the annihilator intersection graph of $L$, denoted by $\mathrm{\text{AIG}}(L).$ The vertex set of $\mathrm{\text{AIG}}(L)$ is the set of all nonzero zero-divisors of $L$ and any two distinct vertices $u$ and $v$ are adjacent in $\mathrm{\text{AIG}}(L)$ if and only if $\mathrm{\text{Ann}}(u)\cap \mathrm{\text{Ann}}(v)\ne \{0\}$. It has shown that the $\mathrm{\text{AIG}}(L)$ is disconnected if and only if the number of atoms in $L$ is two. If $\mathrm{\text{AIG}}(L)$ is connected, then we determine the diameter and the girth of $\mathrm{\text{AIG}}(L).$ We characterize all lattices whose annihilator intersection graph is planar. Further, we obtain the clique number and chromatic number of $\mathrm{\text{AIG}}(L)$ when $L$ is a finite Boolean lattice. We show that the domination number of $\mathrm{\text{AIG}}(L)$ is not exceeding two. Finally, we obtain a condition under which the annihilator intersection graph is identical with the zero-divisor graph and the annihilator ideal graph of lattices.

The zero-divisor graph of a ring $R$ is a graph whose vertex set consists of all the elements of $R$ and distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. We study the zero-divisor graph ${\mathrm{\Gamma }}_0({\mathbb{Z}}_n)$, for $n=p^k,pq,pqr$, where $p,q$ and $r$ are distinct primes. Topological indices such as Wiener and Harary indices of ${\mathrm{\Gamma }}_0({\mathbb{Z}}_n)$ are determined.

The stable equivalent set is a finite collection of disjoint vertex subsets for a connected graph $G=(V(G),E(G))$ such that each set induces the same maximal independent set of $G$ and their union equals $V(G)$. The stable equivalence number is the maximum cardinality of the stable equivalent set $S$, and it is represented by the symbol $S_{\alpha }(G)$. In order to analyze the fault-tolerant and local metric dimensions of the intersection graph of gamma sets in the zero-divisor graph, $I_{\mathrm{\Gamma }}(R)$, the stable equivalent set of $I_{\mathrm{\Gamma }}(R)$ was used.

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 C_G(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 K_p-1, a vertex c ∈ C_G(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 (K_p)).

Let R be an associative ring with two-sided multiplicative identity. In this paper, in the case that R is a commutative α-compatible ring, we compare the diameter (and girth) of the zero-divisor graphs Γ(R) and Γ(R[x;α, δ]). Moreover, we study the zero-divisors of the Öre extension ring R[x;α, δ], whenever R is reversible and (α, δ)-compatible.

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.]

In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z₀(Γ(R)) and T₀(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z₀(Γ(R)) and T₀(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T₀(Γ(R)).

Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom(M, R). We observe that over a commutative ring R, Γ(M) is connected and diam(Γ(M)) ≤ 3. Moreover, if Γ(M) contains a cycle, then gr(Γ(M)) ≤ 4. Furthermore if ∣Γ(M)∣ ≥ 1, then Γ(M) is finite if and only if M is finite. Also if Γ(M) = ∅, then any nonzero f ∈ Hom(M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R₁ × R₂ × ⋯ × R_n, where each R_i is local and this representation of R is unique up to factors R_i with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

This paper is concerned with investigating the center and radius of zero-divisor and ideal-divisor graphs of finite commutative rings. In particular, we examine the interplay between the center and cut-sets of zero-divisor graphs. Of interest are conditions that need to be met in order for either of these sets to contain the other. We then explore the radius and central sets of ideal-divisor graphs, particularly the classification of the center via radius results.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc(G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

In this paper, we extend the concept of the cozero-divisor graph to any arbitrary ring with nonzero identity. Also we study some basic properties of this graph and gain some results on the cozero-divisor graphs of matrix rings.

In this paper, we define and study the annihilator graph associated to a commutative semigroup with zero. Also, we characterize all annihilator graphs with three vertices.

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by Γ_I(_RM) or simply Γ_I(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ₀(R); this is an important result on the definition. We prove that if ann_R(M) ⊆ I and H is the subgraph of Γ_I(M) induced by the set of all non-isolated vertices, then diam(H) ≤ 3 and gr(Γ_I(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec(R) and ω(Γ_Nil(R)(M)) are finite, then χ(Γ_Nil(R)(M)) ≤ ∣Spec(R)∣ + ω(Γ_Nil(R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of Γ_P(M), where ann_R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann_R(M) ⊆ I, Γ_I(M) is a forest if and only if Γ_I(M) is a union of isolated vertices or a star.

Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc(G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc(G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc(G) to |V(G)| and show that there is a finite connected graph G with loc(G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The essential graph of $R$ is defined as the graph $EG(R)$ with the vertex set $Z{(R)}^{*}=Z(R)\setminus \{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if ann${}_R(xy)$ is an essential ideal. It is proved that $EG(R)$ is connected with diameter at most three and with girth at most four, if $EG(R)$ contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

Let ${\mathbb{Z}}_{p^s}$ be the ring of integers modulo $p^s$ where $p$ is a prime and $s(\ge 1)$ is a positive integer, $R=M_{2\times 2}({\mathbb{Z}}_{p^s})$ the $2\times 2$ matrix ring over ${\mathbb{Z}}_{p^s}$. The zero-divisor graph of $R$, written as $\mathrm{\Gamma }(R)$, is a directed graph whose vertices are nonzero zero-divisors of $R$, and there is a directed edge from a vertex $A$ to a vertex $B$ if and only if $AB=0$. In this paper, we completely determine the automorphisms of $\mathrm{\Gamma }(R)$.

For a *-ring $A$, we associate a simple undirected graph ${\mathrm{\Gamma }}^{*}(A)$ having all nonzero left zero-divisors of $A$ as vertices and, two vertices $x$ and $y$ are adjacent if $x{y}^{*}=0$. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having ${\mathrm{\Gamma }}^{*}(A)$ a complete graph or a star graph, and sufficient conditions are obtained for ${\mathrm{\Gamma }}^{*}(A)$ to be connected and also for ${\mathrm{\Gamma }}^{*}(A)$ to be disconnected. For a Rickart *-ring $A$, we characterize the girth of $\mathrm{\text{gr}}({\mathrm{\Gamma }}^{*}(A))$ and prove a sort of Beck’s conjecture.

Let $R$ be a commutative ring with identity and let $Z(R)$ be the set of zero-divisors of $R$. The essential graph of $R$ is defined as the graph $\mathrm{\text{EG}}(R)$ with the vertex set $Z{(R)}^{\ast }=Z(R)\backslash \{0\},$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{\text{ann}}}_R(xy)$ is an essential ideal. In this paper, we classify all finite commutative rings with identity for which the genus of $\mathrm{\text{EG}}(R)$ is two.