Nonzero Intersection Research Articles

Non-existence of quasi-symmetric designs having certain pseudo-geometric block graphs

The block graph of a quasi-symmetric design is strongly regular. It is a challenging problem to decide which strongly regular graphs are block graphs of quasi-symmetric designs. We rule out the possibility of quasi-symmetric designs with non-zero intersection numbers whose block graphs have the parameters as certain pseudo-geometric graphs.

Topological indices of intersection ideal graph of ℤn

Let n be a positive integer. The intersection graph of ring R=ℤn, is a graph whose vertices are non-empty and proper ideals of ℤn and two vertices are adjacent if and only if corresponding ideals have non-zero intersection and is denoted by G(ℤn). In this paper we compute some topological indices of this graph.

Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Let <img src=image/13424240_02.gif> be a commutative ring with unity. The essential ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_03.gif>, is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are adjacent whenever <img src=image/13424240_06.gif> is an essential ideal. An essential ideal <img src=image/13424240_04.gif> of a ring <img src=image/13424240_02.gif> is an ideal <img src=image/13424240_04.gif> of <img src=image/13424240_02.gif> (<img src=image/13424240_07.gif>), having nonzero intersection with every other ideal of <img src=image/13424240_02.gif>. The set <img src=image/13424240_08.gif> contains all the maximal ideals of <img src=image/13424240_02.gif>. The Jacobson radical of <img src=image/13424240_02.gif>, <img src=image/13424240_09.gif>, is the set of intersection of all maximal ideals of <img src=image/13424240_02.gif>. The comaximal ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_11.gif>, is a simple graph with vertices as proper ideals of A not contained in <img src=image/13424240_09.gif> and the vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are associated with an edge whenever <img src=image/13424240_10.gif>. In this paper, we study the structural properties of the graph <img src=image/13424240_03.gif> by using the ring theoretic concepts. We obtain a characterization for <img src=image/13424240_03.gif> to be isomorphic to the comaximal ideal graph <img src=image/13424240_11.gif>. Moreover, we derive the structure theorem of <img src=image/13424240_12.gif> and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of <img src=image/13424240_12.gif> and determine the values of <img src=image/13424240_13.gif> for which <img src=image/13424240_12.gif> is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to <img src=image/13424240_12.gif> for some <img src=image/13424240_13.gif>.

Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of 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 .

A survey on the Intersection graphs of ideals of rings

Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. Thus much research has been conducted last few years to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009.

Strongly K-nonsingular Modules

A submodule N of a module M is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.

Some results on the intersection graph of submodules of a module

Abstract Let R be a ring with identity and M be a unitary left R-module. The intersection graph of submodules of M, denoted by G(M), is defined to be a graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have non-zero intersection. In this paper, we consider the intersection graph of submodules of a module. We determine the structure of modules whose clique numbers are finite. We show that if 1 &lt; ω(G(M)) &lt; ∞, then M is a direct sum of a finite module and a cyclic module, where ω(G(M)) denotes the clique number of G(M). We prove that if ω(G(M)) is not finite, then M contains an infinite clique. Among other results, it is shown that a Noetherian R-module whose intersection of all non-trivial submodules is non-zero, is Artinian.

Study of the Dynamics of a Class of One-dimensional Piecewise Linear Displays with One Gap

In the paper, the dynamics of a class of one-dimensional piecewise linear displays with one gap is studied. Stable conditions of equilibrium as well as other attractors are found by numerical methods. During the investigation two basic cases to which all remaining ones come down are considered. In the space of parameters, the areas responding to these or those phase reorganizations are selected. In particular, it was ascertained that for this class of functions, under condition of a continuity on the considered display, there is no set of parameters of it that in case of the given restrictions on the function there were at least two attractors. In case of the existence of a gap is there are infinitely many areas in which two attracting cycles coexist, and if in the area there are two attracting cycles, their periods differ exactly by a unit, and there are no areas where there would be three or more attractors. Besides, it was revealed that in case of three-dimensional motion of parameters along a straight line steady cycles of the various periods with the following important feature are watched: each area supports exactly one or exactly two attracting cycles, and the area containing \(k\) attracting cycles adjoin to the areas containing \(3 − k\) attracting cycles, and sets of values of the periods of any two adjoining areas have a nonzero intersection.

A PRIME ESSENTIAL RING THAT GENERATES A SPECIAL ATOM

A special atom (respectively, supernilpotent atom) is a minimal element of the lattice $\mathbb{S}$ of all special radicals (respectively, a minimal element of the lattice $\mathbb{K}$ of all supernilpotent radicals). A semiprime ring $R$ is called prime essential if every nonzero prime ideal of $R$ has a nonzero intersection with each nonzero two-sided ideal of $R$. We construct a prime essential ring $R$ such that the smallest supernilpotent radical containing $R$ is not a supernilpotent atom but where the smallest special radical containing $R$ is a special atom. This answers a question put by Puczylowski and Roszkowska.

Semigroup rings and group rings with large center

A ring R is called a ring with a large center or an IIC-ring if any nonzero ideal of R has a nonzero intersection with the center of R. We consider conditions which guarantee that a semigroup ring over an IIC-ring is an IIC-ring.

Some Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings

Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R).

Rings of quotients for rings with big center

A ring R is called an IIC-ring if any nonzero ideal of R has nonzero intersection with the center of R. We consider certain results about rings of quotients of semiprime IIC-rings and show by examples that these properties are not preserved in the case of arbitrary IIC-rings. We also prove more general properties of IIC-rings concerning its rings of quotients.

The Maslov cycle as a Legendre singularity and projection of a wavefront set

A Maslov cycle is a singular variety in the lagrangian grassmannian Λ(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of Λ(V). We show here that S is the wavefront set of a Fourier integral distributionwhich is “evaluation at 0 of the quantizations”.

Associative rings with large center

A ring R is called a ring with large center if any nonzero ideal of R has nonzero intersection with the center of R. We give some conditions for an ideal of a ring with large center to be itself a ring with large center, and also we provide an example of a ring with large center R and its ideal I ◁ R such that I is not a ring with large center.

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

In this paper we prove three theorems about twisted generalized Weyl algebras (TGWAs). First, we show that each non-zero ideal of a TGWA has non-zero intersection with the centralizer of the distinguished subalgebra R. This is analogous to earlier results known to hold for crystalline graded rings. Second, we give necessary and sufficient conditions for the centralizer of R to be commutative (hence maximal commutative), generalizing a result by V. Mazorchuk and L. Turowska. And third, we generalize results by D.A. Jordan and V. Bavula on generalized Weyl algebras by giving necessary and sufficient conditions for certain TGWAs to be simple, in the case when R is commutative. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples. We also discuss how simplicity of a TGWA is related to the maximal commutativity of R and the (non-)existence of non-trivial Zn-invariant ideals of R.

Overlapping correlation clustering

We introduce a new approach for finding overlapping clusters given pairwise similarities of objects. In particular, we relax the problem of correlation clustering by allowing an object to be assigned to more than one cluster. At the core of our approach is an optimization problem in which each data point is mapped to a small set of labels, representing membership in different clusters. The objective is to find a mapping so that the given similarities between objects agree as much as possible with similarities taken over their label sets. The number of labels can vary across objects. To define a similarity between label sets, we consider two measures: (i) a 0–1 function indicating whether the two label sets have non-zero intersection and (ii) the Jaccard coefficient between the two label sets. The algorithm we propose is an iterative local-search method. The definitions of label set similarity give rise to two non-trivial optimization problems, which, for the measures of set-intersection and Jaccard, we solve using a greedy strategy and non-negative least squares, respectively. We also develop a distributed version of our algorithm based on the BSP model and implement it using a Pregel framework. Our algorithm uses as input pairwise similarities of objects and can thus be applied when clustering structured objects for which feature vectors are not available. As a proof of concept, we apply our algorithms on three different and complex application domains: trajectories, amino-acid sequences, and textual documents.

On the Banach ⁎-algebra crossed product associated with a topological dynamical system

Given a topological dynamical system Σ=(X,σ), where X is a compact Hausdorff space and σ a homeomorphism of X, we introduce the Banach ⁎-algebra crossed product ℓ1(Σ) most naturally associated with Σ and initiate its study. It has a richer structure than its well investigated C⁎-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of C(X) in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.

INTERSECTION GRAPH OF SUBMODULES OF A MODULE

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-moduleM, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph.

Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system

If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product Banach $\sp{*}$-algebra $\ell^1(\Sigma)$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the integers. The commutant $C(X)'_1$ of $C(X)$ in $\ell^1(\Sigma)$ is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant $C(X)'_*$ of $C(X)$ in $C^*(\Sigma)$, the enveloping $C^*$-algebra of $\ell^1(\Sigma)$. This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study $C(X)'_1$ and $C(X)'_*$ in detail in the present paper. The maximal ideal space of $C(X)'_1$ is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of $X\times\mathbb{T}$. We show that $C(X)'_1$ is hermitian and semisimple, and that its enveloping $C^*$-algebra is $C(X)'_*$. Furthermore, we establish necessary and sufficient conditions for projections onto $C(X)'_1$ and $C(X)'_*$ to exist, and give explicit formulas for such projections, which we show to be unique. In the appendix, topological results for the periodic points of a homeomorphism of a locally compact Hausdorff space are given.