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The main objective of this article is to study the Wiener index of zero-divisor graph of the ring of integer modulo n . We present a constructed method to calculate the Wiener index of zero-divisor graph of Z n for any positive integer n .

We translate some graph properties of 𝔸𝔾(R) and Γ(R) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of R is an anti fixed-place ideal. (2) Min(R) does not have any isolated point. (3) Rad(𝔸𝔾 (R)) = 3. (4) Rad(Γ(R)) = 3. (5) Γ(R) is triangulated (6) 𝔸𝔾 (R) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dt t (𝔸𝔾 (R)) = |ℬ(R)| and also if in addition |Min(R)| > 2, then dt(𝔸𝔾 (R)) = |ℬ (R)|. Finally, it is shown that dt(𝔸𝔾 (R)) is finite if and only if dt t (𝔸𝔾 (R)) is finite if and only if Min(R) is finite.

Let R be a commutative ring with nonzero identity and let Z R be its set of zero divisors. The zero-divisor graph of R is the graph Γ R with vertex set V Γ R = Z R ∗ , where Z R ∗ = Z R \ 0 , and edge set E Γ R = x , y : x ⋅ y = 0 . One of the basic results for these graphs is that Γ R is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤ p 2 q 2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.

We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.

Let 𝕊 denote the algebra of sedenions and let ΓO(𝕊) denote its orthogonality graph. One can observe that every pair of zero divisors in 𝕊 generates a double hexagon in ΓO(𝕊). The set of vertices of a double hexagon can be extended to a basis of 𝕊 that has a convenient multiplication table. The set of vertices of an arbitrary connected component of ΓO(𝕊) is described, and its diameter is found. Then, the bijection between the connected components of ΓO(𝕊) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.

Several different versions of “factoriality” have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of “unique factorization rings.” Our work generalizes Anderson et al.’s results about “unique factorization” in R[X], Gilmer and Parker’s characterization of factorial monoid domains, and Hardy and Shores’s classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is “restricted cancellative” or satisfies various “(restricted) ideal cancellation laws.”

The reason for this work is to present and study deterministic multiset automata, multiset automata and their languages with membership values in complete residuated lattice without zero divisors. We build up the comparability of deterministic [Formula: see text]-valued multiset finite automaton and [Formula: see text]-valued multiset finite automaton in sense of recognizability of a [Formula: see text]-valued multiset language. Then, we relate multiset regular languages to a given [Formula: see text]-valued multiset regular languages and vice versa. At last, we present the concept of pumping lemma for [Formula: see text]-valued multiset automata theory, which we utilize to give a necessary and sufficient condition for a [Formula: see text]-valued multiset language to be non-constant.

For a commutative ring R with unity ( $$1\ne 0$$ ), the zero-divisor graph of R is a simple graph with vertices as elements of $$Z(R)^{*}=Z(R)\setminus \{ 0 \}$$ , where Z(R) is the set of zero-divisors of R and two distinct vertices are adjacent whenever their product is zero. An algorithm is presented to create a zero-divisor graph for the ring of Gaussian integers modulo $$2^{n}$$ for $$n\ge 1$$ . The zero-divisor graph $$\Gamma (\mathbb {Z}_{2^{n}}[i])$$ can be expressed as a generalized join graph $$G[G_{1}, \dots , G_{j}]$$ , where $$G_{i}$$ is either a complete graph (including loops) or its complement and G is the compressed zero-divisor graph of $$\mathbb {Z}_{2^{2n}}$$ . Next, we show that the number of isomorphisms between the zero-divisor graphs for the ring of Gaussian integers modulo $$2^{n}$$ and the ring of integers modulo $$2^{2n}$$ is equal to $$\prod _{j=1}^{2(n-1)}2^{j}!$$ .

“Unique factorization” was central to the initial development of ideal theory. We update this topic with several new results concerning notions of “unique ideal factorization rings” with zero divis...

In the context of factorization in monoid rings with zero divisors, we study associate relations and the resulting notions of irreducibility and factorization length. Building upon these facts, we ...

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 ...

In this paper right zero divisor graph of ring R, where R is set of all 2×2 matrices over the ring Z3 of integer modulo 3 is constructed. Algorithm is also presented to find vertices and edges of r...

Topological indices are widely used for characterizing molecular graphs, establishing relationships between structure and properties of molecules. In this article we discuss adjacency matrix and three topological indices: Wiener index, Laplacian energy and Zagreb indices of zero-divisor graphs of $$\mathbb {Z}_n$$ . We also provide a MATLAB code for Laplacian energy and Zagreb indices of $$\varGamma (\mathbb {Z}_n)$$ .

In this paper, we describe such finite non-nilpotent ring $$R$$ that the compressed zero-divisor graph $$\Gamma_{\sim}(R)$$ is complete with loops.

The applications of finite commutative ring are useful substances in robotics and programmed geometric, communication theory, and cryptography. In this paper, we study the vertex-based eccentric topological indices of a zero-divisor graphs of commutative ring ℤp2×ℤq, where p and q are primes.

In this paper, it is proved that for λ > 1, an additive map that strongly preserves the set of λ-scrambling matrices over the Boolean semiring B is a bijection. The general form of such a map over any antinegative commutative semiring with identity and without zero divisors is characterized. Bibliography: 20 titles.

The paper introduces the Cayley–Dickson split-sedenion algebra. Exact expressions for the annihilators and orthogonalizers of its zero divisors are obtained, and these results are applied in describing relation graphs of the split-sedenions in terms of their diameters and cliques.

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 𝕑 2 n {\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.

A recent subject of study linking commutative ring theory with graph theory has been the concept of the zero divisor graph of a commutative ring. The zero divisor graph of a commutative ring exhibi...

In this article left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring of integers modulo 2 is constructed. We present algorithm to compute the vertices and e...

Let R be a commutative ring with nonzero identity and let $$\Gamma (R)$$ denote the zero divisor graph of R. In this paper, we describe the signless Laplacian and normalized Laplacian spectrum of the zero divisor graph $$\Gamma (\mathbb {Z}_n)$$, and we determine these spectrums for some values of n. We also characterize the cases that 0 is a signless Laplacian eigenvalue of $$\Gamma (\mathbb {Z}_n)$$. Moreover, we find some bounds for some eigenvalues of the signless Laplacian and normalized Laplacian matrices of $$\Gamma (\mathbb {Z}_n)$$.

Although a good portion of elementary linear algebra concerns itself with matrices over a field such as ℝ or ℂ, many combinatorial problems naturally surface when we instead work with matrices over a finite field. As some recent work has been done in these areas, we turn our attention to the problem of enumerating the square matrices with entries in ℤpk that are diagonalizable over ℤpk. This turns out to be significantly more nontrivial than its finite-field counterpart due to the presence of zero divisors in ℤpk.

Let R be a commutative ring without identity. The zero-divisor graph of R, denoted by $$\varGamma (R),$$ is a graph with vertex set $$Z(R){{\setminus }} \{0\},$$ which is the set of all non-zero zero-divisor elements of R and two vertices x and y are adjacent if and only if $$xy=0.$$ In this paper, we characterize (up to isomorphism) all finite decomposable commutative rings without identity whose zero-divisor graphs are toroidal.

Let R be a finite ring. The zero-divisor graph of R, written as G(R), is a digraph with vertex set Z*(R), the set of all nonzero zero-divisor of R, and for x,y∈Z*(R) (needless distinct) there exist...

We study the compressed zero-divisor graph of a finite associative ring R. In particular, we describe commutative finite rings with compressed zero-divisor graphs of order 2. Moreover, we find all graphs of order 3 that are the compressed zero-divisor graphs of some finite 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 x y = 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.

In Section 1 of the paper, we prove McCoy's property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule $M[G]$ has very few zero-divisors if and only if the $S$-semimodule $M$ does so. The concept of Auslander semimodules are introduced in this section as well. In Section 2, we introduce Ohm-Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. In Section 3, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras., 20 pages, totally revised and extended. A couple of new results added

Let R be an associative ring with identity and Z*(R) be its set of non-zero zero-divisors. The undirected zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the non-zero zero-divisors of R, and where two distinct vertices r and s are adjacent if and only if rs = 0 or sr = 0. The distance between vertices a and b is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the superimum of these distances. In this paper, first we prove some results about Γ(R) of a semi-commutative ring R. Then, for a reversible ring R and a unique product monoid M, we prove 0≤ diam(Γ(R))≤ diam(Γ(R[M]))≤3. We describe all the possibilities for the pair diam(Γ(R)) and diam(Γ(R[M])), strictly in terms of the properties of a ring R, where R is a reversible ring and M is a unique product monoid. Moreover, an example showing the necessity of our assumptions is provided.

For a commutative ring R with unity ($$1\ne 0$$), the zero-divisor graph of R, denoted by $$\varGamma (R)$$, is a simple graph with vertices as elements of R and two distinct vertices are adjacent whenever the product of the vertices is zero. Further, its signed zero-divisor graph is an ordered pair $$\varGamma _{\varSigma }(R):= (\varGamma (R), \sigma )$$, where for an edge xy, $$\sigma (xy)$$ is ‘$$+$$’ if either x or y or both is a nonzero zero-divisor and ‘−’ otherwise. This article aims at gaining a deeper insight into signed zero-divisor graphs by investigating properties such as balancing, clusterability, sign-compatibility, consistency, $$\mathcal {C}$$-sign-compatibility, and $$\mathcal {C}$$-consistency.

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 o...

The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.

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 o...

Let $$\mathcal{O}$$ be an order, that is a commutative ring with 1 whose additive structure is a free $$\mathbb{Z}$$ -module of finite rank. A generalized number system (GNS for short) over $$\mathcal{O}$$ is a pair $$(p, \mathcal{D})$$ where $$p \in \mathcal{O}[x]$$ is monic with constant term p(0) not a zero divisor of $$\mathcal{O}$$ , and where $$\mathcal{D}$$ is a complete residue system modulo p(0) in $$\mathcal{O}$$ containing 0. We say that $$(p, \mathcal{D})$$ is a GNS over $$\mathcal{O}$$ with the finiteness property if all elements of $$\mathcal{O}[x]/(p)$$ have a representative in $$\mathcal{D}[x]$$ (the polynomials with coefficients in $$\mathcal{D}$$ ). Our purpose is to extend several of the results from a previous paper of Pethő and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order $$\mathcal{O}$$ and GNS $$(p, \mathcal{D})$$ over $$\mathcal{O}$$ , the pair $$(p, \mathcal{D})$$ admits the finiteness property. This is closely related to work of Vince on matrix number systems. Let $$\mathcal{F}$$ be a fundamental domain for $$\mathcal{O}{\otimes}_\mathbb{Z} \mathbb{R}/\mathcal{O}\, {\rm and}\, p \in \mathcal{O}[X]$$ a monic polynomial. For $$\alpha \in \mathcal{O}$$ , define $$p_{\alpha}(x) := p(x+\alpha) {\rm and} \mathcal{D}_{\mathcal{F},p(\alpha)} := p(\alpha)\mathcal{F} \bigcap \mathcal{O}$$ . Under mild conditions we show that the pairs $$(p_{\alpha},\mathcal{D}_{\mathcal{F},p(\alpha)})$$ are GNS over $$\mathcal{O}$$ with finiteness property provided $$\alpha \in \mathcal{O}$$ in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that $$(p_{-m},\mathcal{D}_{\mathcal{F},p(-m)})$$ does not have the finiteness property for each large enough positive rational integer m. We obtain important relations between power integral bases of etale orders and GNS over $$\mathbb{Z}$$ . Their proofs depend on some general effective finiteness results of Evertse and Győry on monogenic etale orders.

Let $$f : A \rightarrow B$$ be a ring homomorphism and J be an ideal of B. In this paper, we give a characterization of zero divisors of the amalgamation which is a generalization of Maimani’s and Yassemi’s work (see Maimani and Yassemi in J Pure Appl Algebra 212(1):168–174, 2008). Furthermore, we investigate the transfer of Prufer domain concept to commutative rings with zero divisors in the amalgamation of A with B along J with respect to f (denoted by $$A\bowtie ^fJ),$$ introduced and studied by D’Anna et al. (Commutative algebra and its applications, Walter de Gruyter, Berlin, 2009, J Pure Appl Algebra 214:1633–1641, 2010). Our results recover well known results on duplications. The main applications constist in the construction of new original classes of Prufer rings that are not Gaussian and Prufer rings with weak global dimension strictly greater than 1.

In this paper we classify commutative ring extensions with exactly two non Prufer domain intermediate rings. An initial step involves the description of the commutative ring extensions with only one non Prufer domain intermediate ring. Some generalizations to the context of rings with zero divisors are proved. We also answer a question which was left open in Jarboui and Aljubran (Ric Mat, https://doi.org/10.1007/s11587-020-00500-0 , 2020). More precisely, let $$S = K[y_1]]...[y_t]]$$ be a K-algebra (not necessarily finitely generated over the field K) having Krull dimension $$n \ge 1$$ . Let I be a nonzero proper ideal of S (not necessarily maximal in S) and D be a proper subring of K. We provide necessary and sufficient conditions in order that $$R = D + I$$ is a maximal non-integrally closed subring of S.

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p

Radio antennas use different frequency bands of Electromagnetic (EM) Spectrum for switching signals in the forms of radio waves. Regulatory authorities issue a unique number (unique identifying call sign) to each radio center, that must be used in all transmissions. Each radio center propagates channels to the two nearer radio centers so they must use distinctive numbers to avoid interruption. The task of effectively apportioning channels to transmitters is known as the Channel Assignment (CA) problem. CA Problem is discussed under the topic of graph coloring by mathematicians. The radio number of a graph can be used in many parts of the field communication. In this paper, we determined the radio number of zero-divisor graphs &Gamma, (Zp2&times, Zq2) for p,q prime numbers.

Let [Formula: see text] be a commutative ring, [Formula: see text] a pair of exact zero divisors, and [Formula: see text]. Let [Formula: see text] be a complex of free [Formula: see text]-modules. In this paper we explicitly compute cohomological operators of [Formula: see text] over [Formula: see text] by constructing endomorphisms of [Formula: see text]. We consider some properties of these cohomological operators, as well as provide an example in which these cohomological operators act non-trivially.

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.

We investigate the behavior of (strong) Prufer rings in a general pullback diagram. We also consider three new Prufer conditions introduced by Klingler, Lucas, and Sharma.

Neutrosophic components (NC) under addition and product form different algebraic structures over different intervals. In this paper authors for the first time define the usual product and sum operations on NC. Here four different NC are defined using the four different intervals: (0, 1), [0, 1), (0, 1] and [0, 1]. In the neutrosophic components we assume the truth value or the false value or the indeterminate value to be from the intervals (0, 1) or [0, 1) or (0, 1] or [0, 1]. All the operations defined on these neutrosophic components on the four intervals are symmetric. In all the four cases the NC collection happens to be a semigroup under product. All of them are torsion free semigroups or weakly torsion free semigroups. The NC defined on the interval [0, 1) happens to be a group under addition modulo 1. Further it is proved the NC defined on the interval [0, 1) is an infinite commutative ring under addition modulo 1 and usual product with infinite number of zero divisors and the ring has no unit element. We define multiset NC semigroup using the four intervals. Finally, we define n-multiplicity multiset NC semigroup for finite n and these two structures are semigroups under + modulo 1 and { M ( S ) , + , × } and { n - M ( S ) , + , × } are NC multiset semirings. Several interesting properties are discussed about these structures.

Let R be a commutative ring with zero divisors. It is well known that if R is integrally closed, then R is a Prufer domain if and only if there is an integer n > 1 such that, for all a,b∈R,(a,b)n=(...

This text by pure algebraic reasons outlines why the spectrum of maximal ideals SpecℂA of a countable-dimensional differential ℂ-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ℂ-homomorphism ψM : A → ℂ(M ∈ SpecℂA) and any a ∈ A the Taylor series $$ {\overset{\sim }{\psi}}_M(a)\overset{\mathrm{def}}{=}\sum \limits_{m=0}^{\infty }{\psi}_M\left({a}^{(m)}\right)\frac{z^m}{m!} $$ has nonzero radius of convergence depending on the element a ∈ A.

Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and F is a field, then the group ring F[G] contains no zero divisor and Kaplansky Unit Conjecture states that if G is a to...

We propose a randomized polynomial time algorithm for computing non-trivial zeros of quadratic forms in 4 or more variables over F q ( t ) , where F q is a finite field of odd characteristic. The algorithm is based on a suitable splitting of the form into two forms and finding a common value they both represent. We make use of an effective formula for the number of fixed degree irreducible polynomials in a given residue class. We apply our algorithms for computing a Witt decomposition of a quadratic form, for computing an explicit isometry between quadratic forms and finding zero divisors in quaternion algebras over quadratic extensions of F q ( t ) .

Motivated by [J. Algebraic Geom. 27 (2018), pp. 203–209] and [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 936–939], we show the equality ( [ X ] − [ Y ] ) ⋅ [ A 1 ] = 0 \left ( [ X ] - [ Y ] \right ) \cdot [ \mathbb {A} ^{ 1 } ] = 0 in the Grothendieck ring of varieties, where ( X , Y ) ( X, Y ) is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type G 2 G _{ 2 } .

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).

Let R and S be commutative rings with unity, ?:?→? a ring homomorphism and J an ideal of S. Then the subring ?⋈??:={(?,?(?)+?)∣?∈? and ?∈?} of ?×? is called the amalgamation of R with S along J with respect to f. In this paper we generalize and improve recent results on the computation of the diameter of the zero-divisor graph of amalgamated algebras and obtain new results. In particular, we characterize the diameter of the zero-divisor graph of amalgamated duplication.

In this paper, connectedness is completely characterized for the complements of the zero-divisor graphs of partially ordered sets. These results are applied to annihilating ideal graphs and intersection graphs of submodules, generalizing some of the work that has recently appeared in the literature. Key words: Zero-divisor graph, intersection graph, annihilating ideal graph, 0-distributive poset, 0-modular poset.

This work presents a sample construction of two algebras both with the ideal of relations defined by a finite Grobner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.