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The main purpose of this paper is to study the Laplacian eigenvalues of the extension of the zero divisor graph, Γ e (Z n) , for some particular values of n. We characterize the values of n that give the equality of the spectral radius and the second-smallest eigenvalue of Γ e (Z n). Finding Wiener index of Γ e (Z n) in terms of its Laplacian eigenvalues is another objective of this paper. [ABSTRACT FROM AUTHOR]

Let [Formula: see text] and [Formula: see text] be a partial transformation semigroup on [Formula: see text] Obviously, the empty set [Formula: see text] is a zero element of [Formula: see text] and denoted by 0. Let [Formula: see text] An element [Formula: see text] is a zero divisor of [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] The set [Formula: see text] is called the annihilator of [Formula: see text] in [Formula: see text] It is clear that [Formula: see text] for all [Formula: see text] Let [Formula: see text] be the set of all zero divisors of [Formula: see text] and [Formula: see text] Generally, if [Formula: see text] then there exists [Formula: see text] in which [Formula: see text] In this paper, we construct the annihilator graph [Formula: see text] of [Formula: see text] which is an undirected simple graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] Furthermore, we prove some basic structural properties of [Formula: see text] and determine invariants for [Formula: see text] such as the diameter, girth, clique, domination number, and independence number.

Let G be a finite group and C(G) denote the set of all non-normal non-cyclic subgroups of G. In this paper, the function δc(G) = 1 |G| P H∈C(G) |H| is introduced. In fact, we prove that, if δc(G) ≤ 10/3, then either G ∼=A5, or G is solvable. We also find some examples of finite groups G with δc(G) ≤ 10/3. [ABSTRACT FROM AUTHOR]

Let 픽QG be a finite group algebra. We denote by P(픽QG) the probability that the product of two elements of 픽QG be zero. In this paper, we obtain several results on this probability including a computing formula and characterizations. In particular, the computing formula for the P(픽QG) are established where G is the cyclic group Cn, the Quaternion group Qs, the symmetric group S3 and Fq is a finite field. [ABSTRACT FROM AUTHOR]

This research has broadened the definition of the neutrosophic regular in neutrosophic rings, similar to what is known in classical rings. We have studied the properties of neutrosophic regular elements and the most important properties that link them to the neutrosophic (zero divisor, idempotent, unit and nilpotent) elements in neutrosophic rings, and we reached several important results linking these elements to each other, as some of them are different from what is known in classical rings.

The purpose of this paper is to study the strong version of countably A-rings (or the countably McCoy rings) introduced by T. Lucas in [21]. Moreover, we introduce and investigate the module theoretic version of the strongly countably A-ring notion, namely the strongly countably A-modules. We focus especially on the behavior of the stronlgy countably A-property vis-à-vis the polynomial ring, the power series ring, the idealization and the amalgamated duplication of a ring along an ideal. [ABSTRACT FROM AUTHOR]

This research has broadened the definition of the neutrosophic regular in neutrosophic rings, similar to what is known in classical rings. We have studied the properties of neutrosophic regular elements and the most important properties that link them to the neutrosophic (zero divisor, idempotent, unit and nilpotent) elements in neutrosophic rings, and we reached several important results linking these elements to each other, as some of them are different from what is known in classical rings. The most important of which are: If R(I) is a neutrosophic right (left) strongly regular neutrosophic ring, then NSNilR(I) = {0}. In any neutrosophic field R(I) is achieved: R(I) = NRegR(I), although there are some elements that are not neutrosophic unit, NUR(I) n NZR(I) = {bI, b ∈ UR}, and NU R(I) n NIdR(I) = {1, l}. [ABSTRACT FROM AUTHOR]

Vibration performance is the one of the important performances in automotive product because of related to not only ride comfort and control performance of vehicle but also durability and noise performance. However, it is difficult to design efficiently because the performance is determined by the influence of all components constituting of a whole structure. In previous paper, we presented an effective way to design vibration performance based on an inverse method for structural modification that keeps one specified eigenfrequency and its eigen vector of the whole structure the same. In this paper, we propose a new method to keep the specified multiple eigenfrequencies and their eigen vectors the same. Firstly, the matrices for structural modification are calculated as dynamic stiffness matrix variation from the zero-divisors of the specified multiple eigen vectors. The design of dynamic stiffness matrix variation is simplified by representing the matrix variation as just stiffness matrix variation which is independence on eigen frequency. Furthermore, it is possible to represent the calculated matrices redundantly by using an arbitrary non-zero weighting matrix. However, the matrices are generally calculated as fully populated and non-symmetric matrices. Therefore, secondly, we transform the matrices calculated as the zero-divisors into the sparse matrix of reduced row echelon form in advance. This transformation simplifies building symmetric and sparse matrices for realizable structural modification. This sparse matrix, redundantly represented by a weighting matrix, allows a value analysis to select among alternative structural changes to simplify structural complexity while keeping several performances originated from vibration performance. Finally, the proposed method was applied to a numerical case study.

In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Z p for special values of prime integer p. [ABSTRACT FROM AUTHOR]

For a weight function in the unit disk which is the modulus of a finite product of powers of Blaschke factors, we give a canonical representation for the reproducing kernel of the corresponding weighted Bergman space in terms of the values of the kernel and its derivatives at the origin. This yields a formula for the contractive zero divisor of a Bergman space corresponding to a finite zero set. [ABSTRACT FROM AUTHOR]

The set of zero divisors of an MV-algebra is investigated. It is proved that the set of all zero divisors of an MV-algebra A is the union of all prime ideals of A. Next, Z°-ideals and Z-ideals are introduced and their properties are studied. We examine the relationship between them and their relationship with the minimal prime ideals. We study their behavior under homomorphism of MV-algebras. Also, we investigate in what conditions the annihilator of an ideal is a Z°-ideal and Z-ideal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new kinds of graphs based on equality algebras. First of all, by using the meet operation we define the notion of zero divisors on equality algebra and study related properties. Then we introduce a meet graph on equality algebra by using zero divisors. In addition, we investigate some conditions for a meet graph to be a complete, connected and a star graph. Also, by using the notion of filters in equality algebras, we define an equivalence relation on equality algebras and then by using the equivalence classes we introduce two kinds of graphs on them. Finally, conditions for a graph based on these classes to be connected, or bipartite, or complete, or planar or an outer-planar graph are provided. [ABSTRACT FROM AUTHOR]

NV (Noise and Vibration) performance is determined by the influence of all components constituting a whole structure. It is difficult to design NV performance efficiently because the structural modification of a certain component affects the performance of the whole structure. As one of the effective ways to design the performance, this paper presents an inverse method for structural modification that keeps the specified eigenfrequency and its modal vector of the whole structure the same. Firstly, the matrices for structural modification are calculated as dynamic stiffness matrix variation from the zero-divisors of a specified modal vector. Furthermore, it is possible to represent the calculated matrices redundantly by using an arbitrary non-zero weighting matrix. This makes it possible to obtain various solutions for structural modification. However, the matrices are generally calculated as fully populated and non-symmetric matrices. Therefore, with these matrices, it is difficult to feasibly find a symmetrical and sparse mass or stiffness matrix to use for structural modifications in applications that focus on designing specified regions of the whole structure. Secondly, we propose how to transform the matrices as the zero-divisors into the sparse matrix of reduced row echelon form in advance. This transformation simplifies building symmetric and sparse matrices for realizable structural modification. The sparse matrix, redundantly represented by a weighting matrix, allows a value analysis to select among alternative structural changes to lighten the product or simplify its complexity. Finally, the proposed method was applied to a numerical case study.

Let R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) < 1/4 if and only if RG Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4. [ABSTRACT FROM AUTHOR]

In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying algebraic operations on fuzzy posets and bounded fuzzy lattices. We first prove that fuzzy posets are closed under finite direct products whenever the triangular norm realizing the product construction has no zero divisors. This result is then extended to the case of bounded fuzzy lattices. Some immediate consequences are then obtained within the setting of direct products realized by triangular norms with no nilpotent elements as well as strictly monotone and cancellative triangular norms. We then introduce a triangular norm based construction of ordinal products and similarly show that fuzzy posets are closed under ordinal products whenever the triangular norm realizing the product construction has no zero divisors. [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p.

Let R be a finite ring and let ${\mathrm {zp}}(R)$ denote the nullity degree of R, that is, the probability that the multiplication of two randomly chosen elements of R is zero. We establish the nullity degree of a semisimple ring and find upper and lower bounds for the nullity degree in the general case. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. The regular digraph of ideals of R, denoted by Γ(R), is a digraph whose vertex-set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J, whenever I contains a non-zero zero divisor on J. We generalize this notion with respect to an ideal I0 of R and denote it by ...(R) in such a way that I0 = 0 gives us Γ(R). Also by verifying connectedness and diameter of ...(R) we will observe that there is a strong relation between ...(R) and Γ(R)/I0). Finally, we consider the number of non-singular connected components of ...(R). [ABSTRACT FROM AUTHOR]

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x 2 + b x + c = 0 in split quaternion algebra. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce and study the notion of Property (A) of a ring R or an R-module M with respect to an ideal I of R. The new notion turns out to be a weak form of the classic notion of an A-ring, in the sense that, any R-module satisfying the Property (A) satisfies as well the Property (A) with respect to any ideal I of R. Moreover, we prove that if I is contained in the nilradical of R, then the notion of A-module with respect to I and the notion of A-module collapse. Also, we present an example of a ring R possessing an ideal I ⊆ Z(R) such that R is an A-ring with respect to I while R is not an A-ring. Finally, we totally characterize the rings R and the R-modules M satisfying the Proprety (A) with respect to an ideal I as well as we investigate the behavior of the Property (A) with respect to an ideal vis-à-vis the direct products of rings and modules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R≅Π𝕑2n{\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

The main purpose of this paper is to totally characterize when the amalgamated duplication R ⋈ I of a ring R along an ideal I is an -ring as well as an -ring. In this regard, we prove that R ⋈ I is an -ring if and only if R is an -ring and I is contained in the set of zero divisors Z(R) of R. As to the Property () of R ⋈ I, it turns out that its characterization involves a new concept that we introduce in [6] and that we term the Property () of a module M along an ideal I. In fact, we prove that R ⋈ I is an -ring if and only if R is an -ring, I is an -module along itself and if p is a prime ideal of R such that p ⊆ ZR(I) ∪ Z1 (R), then either p ⊆ ZR(I) or p ⊆ Z1 (R), where Z1 (R) := {a ∈ R : a + I ⊆ Z1z(R)}. [ABSTRACT FROM AUTHOR]

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study some special elements (zero divisors and units) in ternary rings. Then, the main properties and concepts of noetherian and artinian ternary rings have been studied. In addition, new results on noetherian and artinian ternary rings have been investigated. [ABSTRACT FROM AUTHOR]

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.

In this paper we show that if {Ri}i∈I is a family of commutative rings and {Mi}i∈I is a family of modules such that each Mi is an Ri-module, then the direct product Mi is a -module over Ri if and only if each Mi is an -module over Ri, i ∈ I. This result extends the work of Hong, Kim, Lee and Ryu in Rings with Property () and their extensions, J. Algebra315 (2007), 612–628. Moreover, we characterize Property () for modules in terms of Property () of their total quotient modules, extending the work of Dobbs and Shapiro in On the strong ()-ring of Mahdou and Hassani, Mediterr. J. Math.10 (2013), 1995–1997. [ABSTRACT FROM AUTHOR]

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras.Arecent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length 2 cannot be happen. The first unsettled case is the existence of zero divisors of length 3. Here we study possible length 3 zero divisors in the rational group algebras and in the group algebras over the field Fp with p elements for some prime p. As a consequencewe prove that the rational group algebras of torsion-free groups which are residually finite p-group for some prime p ≠ 3 have no zero divisor of length 3. We note that the determination of all zero divisors of length 3 in group algebras over F2 of cyclic groups is equivalent to find all trinomials (polynomials with 3 non-zero terms) divided by irreducible polynomials over F2. The latter is a subject studied in coding theory and we add here some results, e.g. we show that 1 + x + x² is a zero divisor in the group algebra over F2 for some element x of the group if and only if x is of finite order divided by 3 and we find all β in the group algebra of the shortest length such that (1 + x + x²)β = 0; and 1 + x² + x³ or 1 + x + x³ is a zero divisor in the group algebra over F2 for some element x of the group if and only if x is of finite order divided by 7. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and investigate a new class of rings lying properly between the class of 𝒜 -rings and the class of 𝒮 𝒜 -rings. The new class of rings, termed the class of 𝒫 𝒮 𝒜 -rings, turns out to share common characteristics with both 𝒜 -rings and 𝒮 𝒜 -rings. Numerous properties and characterizations of this class are given as well as the module-theoretic version of 𝒫 𝒮 𝒜 -rings is introduced and studied. [ABSTRACT FROM AUTHOR]

We present an answer to a problem raised by Anderson and Chun in [D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16(7) (2017) 1750143] on characterizing when the idealization R ⋉ M of a ring R on an R -module M is an 𝒜 -ring (respectively, an 𝒮 𝒜 -ring) in terms of module-theoretic properties of R and M. Also, we are concerned with an open question asked by these two authors which reads the following: What modules over a given ring R are homomorphic images of modules satisfying the strong Property 𝒜 ? (see, Question 4.4(1) in the above mentioned paper). This paper highly contributes to answer such a question. [ABSTRACT FROM AUTHOR]

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization R ⋉ M is an A -ring if and only if R is an A -ring [14, Proposition 3.5]. In effect, we drop the reduceness hypotheses and prove that, given an arbitrary commutative ring R and any submodule M of a flat R-module F, R ⋉ M is an A -ring (resp., S A -ring) if and only if R is an A -ring (resp., S A -ring). [ABSTRACT FROM AUTHOR]

We study Pauli's theorem in Clifford algebras. We offer an algorithm for constructing the Pauli operator and show that the problem of constructing the Pauli operator is connected with that of zero divisors in Clifford algebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M, denoted by G(M), is an undirected simple graph whose vertices are the elements of ZR(M)\AnnR(M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of G(M). We show that the zero-divisor graph of M has a universal vertex in ZR(M)rr(AnnR(M)) if and only if R = ⊕Z2⊕R' and M = Z2⊕M', where M' is an R'-module. Moreover, we show that if G(M) is a complete graph, then one of the following statements is true: (i) AssR(M) = {m1,m2}, where m1,m2 are maximal ideals of R. (ii) AssR(M) = {p}, where p²⊆ AnnR(M). (iii) AssR(M) = {p}, where p³⊆AnnR(M). [ABSTRACT FROM AUTHOR]

We introduce, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true over some relevant collections of instances and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test within the dynamic geometry program GeoGebra. Our framework has some significant differences regarding some alternative, recent formulation of the "true on components" idea; differences and similarities between both approaches are discussed here. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with nonzero identity. The nullity degree of R , denoted by z p (R) , is the probability that the multiplication of two randomly chosen elements of R is zero. In this paper, we characterize all rings R with z p (R) ≥ 1 4 . Also, we study rings R with z p (R) ≥ 1 p 2 , where p is a prime number. [ABSTRACT FROM AUTHOR]