Zero Divisors Research Articles

Total and Cototal Domination Number of Some Zero Divisor Graph

The concept of zero divisor graphs of a commutative ring leads to various research topics since it relates both Ring theory and Graph theory. Domination in graphs has its own development in each field it enters. In this paper, We have given few results on Total and cototal domination number of some zero divisor graph on direct product of commutative rings.

Zero Divisor Graph of a Lattice and Its Unique Ideal

Let L be a lattice with the least element 0. Let AL be the finite set of atoms with AL>1 and ΓL be the zero divisor graph of a lattice L. In this paper, we introduce the smallest finite, distributive, and uniquely complemented ideal B of a lattice L having the same number of atoms as that of L and study the properties of ΓL and ΓB.

The universal enveloping algebra of and the Racah algebra

Let denote a field with The Racah algebra is the unital associative -algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert thatand each of the elementsis central in Additionally the element is central in In this paper, we explore the relationship between the Racah algebra and the universal enveloping algebra Let a, b, c denote mutually commuting indeterminates. We show that there exists a unique -algebra homomorphism that sendswhere x, y, z are the equitable generators for We additionally give the images of and certain Casimir elements of under We also show that the map is an injection and thus provides an embedding of into We use the injection to show that contains no zero divisors.

Classification of non-local rings with genus two zero-divisor graphs

The zero-divisor graph of a commutative ring R is a simple graph whose vertices are the nonzero zero divisors of R and two distinct vertices are adjacent if their product is zero. In this article, we determine precisely all non-local commutative rings whose zero-divisor graphs have genus two.

On Eccentric Topological Indices Based on Edges of Zero Divisor Graphs

This article is devoted to the determination of edge-based eccentric topological indices of a zero divisor graph of some algebraic structures. In particular, we computed the first Zagreb eccentricity index, third Zagreb eccentricity index, geometric-arithmetic eccentricity index, atom-bond connectivity eccentricity index and a fourth type of eccentric harmonic index for zero divisor graphs associated with a class of finite commutative rings.

Real-normalized differentials: limits on stable curves

We study the behaviour of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We describe all possible limits of RN differentials on any stable curve. In particular we prove that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. We further show that the limits of zeros of RN differentials are the divisor of zeros of a twisted differential — an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials (in this paper, RN differentials, but the method is more general) on smooth Riemann surfaces, in a plumbing neighbourhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighbourhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduced to a construction of a suitable normalized holomorphic differential with prescribed ‘jumps’ (mismatches) along the identified circles (seams). We solve this additive analogue of the multiplicative Riemann–Hilbert problem in a new way, by using iteratively the Cauchy integration kernels on the irreducible components of the stable curve, instead of using the Cauchy kernel on the plumbed smooth surface. As the stable curve is fixed, this provides explicit estimates for the differential constructed, and allows a precise degeneration analysis. Bibliography: 22 titles.

Upper dimension and bases of zero-divisor graphs of commutative rings

For a commutative ring R with non-zero zero divisor set Z∗(R), the zero divisor graph of R is Γ(R) with vertex set Z∗(R), where two distinct vertices x and y are adjacent if and only if xy=0. The upper dimension and the resolving number of a zero divisor graph Γ(R) of some rings are determined. We provide certain classes of rings which have the same upper dimension and metric dimension and give an example of a ring for which these values do not coincide. Further, we obtain some bounds for the upper dimension in zero divisor graphs of commutative rings and provide a subset of vertices which cannot be excluded from any resolving set.

Zero divisor and unit elements with supports of size 4 in group algebras of torsion-free groups

Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and is a field, then the group ring contains no zero divisor and Kaplansky Unit Conjecture states that if G is a torsion-free group and is a field, then contains no non-trivial units. The support of an element in , denoted by , is the set . In this paper, we study possible zero divisors and units with supports of size 4 in group algebras of torsion-free groups. We prove that if α, β are non-zero elements in for a possible torsion-free group G and an arbitrary field such that and , then . In [J. Group Theory, 16 no. 5, 667–693], it is proved that if is the field with two elements, G is a torsion-free group and such that and , then . We improve the latter result to . Also, concerning the Unit Conjecture, we prove that if for some and , then .

On the adjacency matrices of the Anderson-Livingston zero divisor graphs of Galois rings

On the adjacency matrices of the Anderson-Livingston zero divisor graphs of Galois rings

Construction Algorithm for Zero Divisor Graphs of Finite Commutative Rings and Their Vertex-Based Eccentric Topological Indices

Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this paper we discuss eccentric topological indices of zero divisor graphs of commutative rings Z p 1 p 2 × Z q , where p 1 , p 2 , and q are primes. To enhance the importance of these indices a construction algorithm is also devised for zero divisor graphs of commutative rings Z p 1 p 2 × Z q .

Computing metric dimension of compressed zero divisor graphs associated to rings

Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).

Adaptive Gaussian mixture filter for Markovian jump nonlinear systems with colored measurement noises

This paper considers the state estimation of discrete-time Markovian jump nonlinear systems with colored measurement noises obeying a nonlinear autoregressive process of order n, which is motivated by tracking the maneuvering target under electronic countermeasures with high speed sampling or persistent perturbations. In order to remove the measurement noises correlation, the left zero divisor is explored to reconstruct a new measurement equation via difference approach, with the help of applying statistical linear regression to the colored measurement noise model. Then, a novel hypothesis set constituted of all possible values of multi-step Markov jumping parameters is defined and the posterior probability density of the state is derived recursively. By using Gaussian mixtures to approximate the posterior probability densities, an adaptive Gaussian mixture filter for the considered system is proposed, where the Gaussian components with small weights are pruned adaptively through measuring the Alpha (or Beta) divergence for the original and approximated Gaussian mixtures, to achieve a tradeoff between the estimation accuracy and running time. A maneuvering target tracking accompanied by range gate pull-off with different colored measurement noises cases is simulated to validate the proposed method.

On some properties of KP-II soliton divisors in $$Gr^\mathrm{TP}(2,4)$$ G r TP ( 2 , 4 )

We construct the pole divisor of the wavefunction for real regular bounded multi-soliton KP-II solutions represented by points in $$Gr^\mathrm{TP} (2,4)$$ on the reducible rational $$\mathtt M$$ -curve $$\varGamma ({\mathcal {N}}_T)$$ recently introduced in Abenda and Grinevich (KP theory, plane-bipartite networks in the disk and rational degenerations of $${\mathtt M}$$ -curves, 2018. arXiv:1801.00208 ) and we give evidence that the asymptotic behavior of its zero divisor in the real (x, y)-plane at fixed time t is compatible with the behavior of the soliton solution classified in Chakravarty and Kodama (Stud Appl Math 123:83–151, 2009).

On generalized power series rings with some restrictions on zero-divisors

Let [Formula: see text] be a ring and [Formula: see text] a strictly ordered monoid. The construction of generalized power series ring [Formula: see text] generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev–Neumann construction. In this paper, for a reversible right Noetherian ring [Formula: see text] and a m.a.n.u.p. monoid [Formula: see text], it is shown that (i) [Formula: see text] is power-serieswise [Formula: see text]-McCoy, (ii) [Formula: see text] have Property (A), (iii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip, (iv) [Formula: see text] is strongly AB if and only if [Formula: see text] is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring [Formula: see text] and the graph-theoretical properties of its undirected zero divisor graph of [Formula: see text]. A complete characterization for the possible diameters [Formula: see text] is given exclusively in terms of the ideals of [Formula: see text]. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.

A generalization of zero divisor graphs associated to commutative rings

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by $$\Gamma ^n_R$$ , with $$R^n\backslash \{0\}$$ as the vertex set and two distinct vertices X and Y in $$R^n$$ being adjacent if and only if there exists an $$n\times n$$ lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that $$X^TAY=0$$ or $$Y^TAX=0$$ , where, for a matrix $$B, B^T$$ is the matrix transpose of B. If $$n=1$$ , then $$\Gamma ^n_R$$ is isomorphic to the zero divisor graph $$\Gamma (R)$$ , and so $$\Gamma ^n_R$$ is a generalization of $$\Gamma (R)$$ which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of $$\Gamma ^n_ R$$ . We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

Zero divisor graphs of commutative graded rings

We study a natural generalization of the zero divisor graph introduced by Anderson and Livingston to commutative rings graded by abelian groups, considering only homogeneous zero divisors. We develop a basic theory for graded zero divisor graphs and present many examples. Finally, we examine classes of graphs that are realizable as graded zero divisor graphs and close with some open questions.

English

English

Zero divisors and units with small supports in group algebras of torsion-free groups

ABSTRACTKaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field 𝔽, the group ring 𝔽[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in 𝔽[G] whose supports have size 3. For any field 𝔽 and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥10. If 𝔽 = 𝔽2 is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields.

Quotient Hyper BCK-algebra and its Zero Divisor Graph

Quotient Hyper BCK-algebra and its Zero Divisor Graph

Identities, approximate identities and topological divisors of zero in Banach algebras

In [3] S.J. Bhatt and H.V. Dedania exposed certain classes of Banach algebras in which every element is a topological divisor of zero. We identify a new (large) class of Banach algebras with the aforementioned property, namely, the class of non-unital Banach algebras which admits either an approximate identity or approximate units. This also leads to improvements of results by R.J. Loy and J. Wichmann, respectively. If we observe that every single example that appears in [3] belongs to the class identified in the current paper, and, moreover, that many of them are classical examples of Banach algebras with this property, then it is tempting to conjecture that the classes exposed in [3] must be contained in the class that we have identified here. However, we show somewhat elusive counterexamples. Furthermore, we investigate the role completeness plays in the results and show, by giving a suitable example, that the assumptions are not superfluous. The ideas considered here also yields a pleasing characterization: The socle of a semisimple Banach algebra is infinite-dimensional if and only if every socle-element is a topological divisor of zero in the socle.