Left Zero Divisors Research Articles

On fusible rings and related notions

A ring is called left fusible if each of its nonzero elements is the sum of a left zero-divisor and a non-left zero-divisor. This paper aims to extend the existing theory of left fusible rings and related notions such as being left unit-fusible, being regular left fusible, and being uniquely left fusible. New examples of these rings are presented. A new notion of left generalized unit-fusible rings is introduced and discussed. The asymmetry of these notions is addressed.

On fusible rings

A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor, and R is called uniquely left fusible if for any there exists a unique left zero-divisor z such that a – z is non-left zero-divisor. We show that a left fusible ring R is uniquely left fusible if and only if either R is a domain or R has a unique non-left zero-divisor element.

Zero-Divisor Graphs of Order-Decreasing Full Transformation Semigroups

Let 𝑛 ∈ ℤ+ and 𝑋𝑛 = {1,2, … , 𝑛} be a finite set. Let 𝐷𝑛 be the order-decreasing full transformation semigroup on 𝑋𝑛. In this paper, we find the left zero-divisors, the right zero-divisors and two sided zerodivisors of 𝐷𝑛. Moreover, for 𝑛 ≥ 4 we define an undirected graph Γ(𝐷𝑛) whose vertices are two-sided zero divisors of 𝐷𝑛 excluding the zero element 𝜃 of 𝐷𝑛. In the graph, distinct two vertices 𝛼 and 𝛽 are joined if and only if 𝛼𝛽 = 𝜃 = 𝛽𝛼. In this paper, we prove that Γ(𝐷𝑛) is a connected graph, and we find diameter, girth, the degrees of all vertices, the maximum degree and the minimum degree in Γ(𝐷𝑛). Moreover, we give lower bounds for clique number and choromatic number of Γ(𝐷𝑛).

Zero-divisor graphs of partial transformation semigroups

Let $\mathcal P_{n}$ be the partial transformation semigroup on $X_{n}=\{1,2,\ldots ,n\}$. In this paper, we find the left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal P_{n}$, and their numbers. For $n \geq 3$, we define an undirected graph $\Gamma(\mathcal P_{n})$ associated with $\mathcal P_{n}$ whose vertices are the two sided zero-divisors of $\mathcal P_{n}$ excluding the zero element $\theta$ of $\mathcal P_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta =\beta\alpha$. First, we prove that $\Gamma(\mathcal P_{n})$ is a connected graph, and find the diameter, girth, domination number and the degrees of the all vertices of $\Gamma(\mathcal P_{n})$. Furthermore, we give lower bounds for clique number and chromatic number of $\Gamma(\mathcal P_{n})$.

Zero-divisor graphs of Catalan monoid

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal C_{n}$; and their numbers. For $n \geq 4$, we define an undirected graph $\Gamma(\mathcal C_{n})$ associated with $\mathcal C_{n}$ whose vertices are the two sided zero-divisors of $\mathcal C_{n}$ excluding the zero element $\theta$ of $\mathcal C_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta=\beta\alpha$. Then we first prove that $\Gamma(\mathcal C_{n})$ is a connected graph, and then we find the diameter, radius, girth, domination number, clique number and chromatic numbers and the degrees of all vertices of $\Gamma(\mathcal C_{n})$. Moreover, we prove that $\Gamma(\mathcal C_{n})$ is a chordal graph, and so a perfect graph.

A study of right zero divisor graph of 3 × 3 matrices over ℤ2

In [10] we have computed left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring ℤ2 of integers modulo 2 is constructed. We have extended our result for right zero divisor graph of the ring R. Thus we have listed vertices of RZDG by means of lexicographic order. Which enables us to present algorithm that computes the edges of this graph. In addition, it helps to analyze some of its important properties.

INCREASING THE ROUGHNESS OF THE PROCEDURE FOR EVALUATING A VECTOR OF THE CONDITION OF OBJECTS TO THE INFLUENCE OF UNCERTAINTY FACTORS

Algorithms for increasing the roughness of the procedure for assessing the state vector of control objects to the influence of uncertainty factors are given. Expressions are obtained for extended state vectors and observations. Stable inversion algorithms are given for a nondegenerate block matrix with the allocation of its left and right zero divisors of maximum rank. The presented stable computational procedures allow us to regularize the problem of synthesis of algorithms for estimating the parameters of regulators in adaptive control systems with a customizable model and to improve the quality indicators of control processes under conditions of parametric uncertainty

Fixing number and metric dimension of a zero-divisor graph associated with a ring

Let R be a ring and , where and are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by , is a simple undirected graph with vertex set , and two distinct vertices are adjacent if and only if ab = 0 or ba = 0. Let , the ring of integers modulo n, and the ring of all matrices over a finite field of q elements. In this article, using the technique on characteristic matrices, we give the value of fixing number of . Moreover, we calculate the fixing number and metric dimension of and .

Rings in which every element is the sum of a left zero-divisor and an idempotent

Rings in which every element is the sum of a left zero-divisor and an idempotent

Rings in which every left zero-divisor is also a right zero-divisor and conversely

A ring [Formula: see text] is called eversible if every left zero-divisor in [Formula: see text] is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that [Formula: see text] is eversible if and only if its upper triangular matrix ring [Formula: see text] is eversible, and if [Formula: see text] is eversible then [Formula: see text] is eversible.

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.

The zero-divisor graph of a ring with involution

For a *-ring [Formula: see text], we associate a simple undirected graph [Formula: see text] having all nonzero left zero-divisors of [Formula: see text] as vertices and, two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having [Formula: see text] a complete graph or a star graph, and sufficient conditions are obtained for [Formula: see text] to be connected and also for [Formula: see text] to be disconnected. For a Rickart *-ring [Formula: see text], we characterize the girth of [Formula: see text] and prove a sort of Beck’s conjecture.

On total directed graphs of non-commutative rings

For a non-commutative ring R, the left total directed graph of R is a directed graph with vertex set as R and for the vertices x and y, x is adjacent to y if and only if there is a non-zero r∈R which is different from x and y, such that rx+yr is a left zero-divisor of R. In this paper, we discuss some very basic results of left (as well as right) total directed graph of R. We also study the coloring of left total directed graph of R directed graph.

Answers to Some Questions Concerning Rings with Property (A)

AbstractA ring R has right property (A) whenever a finitely generated two-sided ideal of R consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property (A), raised by Hong et al. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if R is a right duo ring and M is a u.p.-monoid (unique product monoid), then R is right M-McCoy and the monoid ring R[M] has right property (A). The second question has a negative answer and we demonstrate this by constructing a suitable example.

Fusible rings

ABSTRACTA ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that if R is a left fusible ring and σ is a ring automorphism of R, then R[x;σ] and R[[x;σ]] are left fusible. It is proved that if R is a left fusible ring, then Mn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semisimple clean rings. A ring R is called left unit fusible if every nonzero element of R can be written as the sum of a unit and a left zero-divisor in R. Full rings of continuous functions are fusible. It is also shown that if in a ring R, where the ei are orthogonal idempotents and each is left unit fusible, then R is left unit fusible. Finally, we give some properties of fusible rings.

Algebras of cubic matrices

We consider algebras of -cubic matrices (with ). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size m. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff . We introduce a notion of accompanying algebra (which is -dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed.

On Ore extension and skew power series rings with some restrictions on zero-divisors

The study of rings with right Property ([Formula: see text]), has done an important role in noncommutative ring theory. Following literature, a ring [Formula: see text] has right Property ([Formula: see text]) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ([Formula: see text]) of Ore extensions as well as skew power series rings. We will show that if [Formula: see text] is a right duo ring, then the skew power series ring [Formula: see text] has right Property ([Formula: see text]), when [Formula: see text] is right Noetherian and [Formula: see text]-compatible. Moreover, for a right duo ring [Formula: see text] which is [Formula: see text]-compatible, it is shown that (i) the Ore extension ring [Formula: see text] has right Property ([Formula: see text]) and (ii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip. As a corollary of our results, we provide answers to some open questions related to Property [Formula: see text], raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ([Formula: see text]) and their extensions, J. Algebra 315 (2007) 612–628].

Solvability conditions of problem of synthesizing proportional-integral control of a linear MIMO system

The solvability conditions and just the solution of the problem of the regular and irregular proportional-integral (PI) control are found in accordance with the properties of invariant zeros of a multi-input multioutput (MIMO) system. It is proved that the problem of synthesizing the control of the MIMO system is solvable if and only if the pair of matrices (A, B) that describes a control plant is controllable and the matrix BL?ACR? (where BL? is the left zero divisor of the matrix B and CR? is the right zero divisor of the output matrix C) has a complete row rank.

Commutativity of Near-rings with Derivations

In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained derivation d satisfying the condition that d(a) is not a left zero-divisor in R for some a ∈ R. As consequences, we generalize several commutativity theorems for 3-prime near-rings admitting derivations.

Diameter of the zero divisor graph of semiring of matrices over Boolean semiring

Let S be a semiring and let Z(S) be its set of nonzero zero divisors. We denote the zero divisor graph of S by ( S) whose vertex set is Z(S) and there is an edge between the vertices x and y (x6 y) in ( S) if and only if either xy = 0 or yx = 0. In this paper we study the zero divisor graph of the semiring of matrices Mn(B), (n > 1) over the Boolean semiring B. We investigate the properties of the right zero divisors and the left zero divisors of Mn(B) and then use these results to prove that the diameter of ( Mn(B)) is 3.