Quaternion Rings Research Articles

Lie triple derivation and Lie bi-derivation on quaternion rings

In this study, we prove the existence of the central Lie bi-derivation for the ring with identity on the quaternion ring. We also describe the triple Lie derivation using the Jordan derivation on the aforementioned ring. An example is provided to demonstrate that our result is theoretically viable.

Minimal rings related to generalized quaternion rings

The family of rings of the form \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left \langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \right \rangle} is investigated which contains the generalized Hamilton quaternions over $\Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $\Z_{p^s}$ are characterized.

BIPARTITE GRAPH ASSOCIATED WITH ELEMENTS AND COSETS OF SUBRINGS OF FINITE RINGS

Let be a finite ring. The bipartite graph associated to elements and cosets of subrings of is a simple undirected graph with vertex set , where is the set of all subrings of , and two vertices and are adjacent if and only if In this study, we investigate some basic properties of the graph . In particular, we investigate some properties of , where is the ring of matrices over Also, we study the diameter of the bipartite graph associated to the quaternion ring

Applying Quaternions to Recognize Hidden Details in Images: Rothko as a Case Study

Images or paintings with homogeneous colors may appear dull to the naked eye; however, there may be numerous details in the image that are expressed through subtle changes in color. This manuscript introduces a novel approach that can uncover these concealed details via a transformation that increases the distance between adjacent pixels, ultimately leading to a newly modified version of the input image. We chose the artworks of Mark Rothko—famous for their simplicity and limited color palette—as a case study. Our approach offers a different perspective, leading to the discovery of either accidental or deliberate clusters of colors. Our method is based on the quaternion ring, wherein a suitable multiplication can be used to boost the color difference between neighboring pixels, thereby unveiling new details in the image. The quality of the transformation between the original image and the resultant versions can be measured by the ratio between the number of connected components in the original image (m) and the number of connected components in the output versions (n), which usually satisfies nm≫1. Although this procedure has been employed as a case study for artworks, it can be applied to any type of image with a similar simplicity and limited color palette.

Geometrical, thermochemistry and electronic properties of hydrogenated germanium clusters (Ge m H n , m = 3–4, n = 0–10) and their anions

ABSTRACT Based on the density functional theory, Δ2 E(n), De, E gap and EA of hydrogen germanide clusters were studied and discussed. The selective hydrogen absorption of the isomers and anions of Ge m H n (m = 3–4, n = 0–10) was investigated. The relative stability, electrical properties, optimal geometry of Ge x H y clusters and their anions were systematically analysed. The results showed that the growth rule of Ge m H n (m = 3–4, n = 0–10) clusters was hydrogenation. The structure of quaternion ring expanded gradually and finally formed chain structures. The stability of clusters with even hydrogen atom was higher than that with odd hydrogen atom. The results showed that the most stable structure in Ge4H n (n = 0–10) cluster was Ge4H10. With the increasing number of hydrogen atoms, the stable structure of the cluster eventually tended to be chain-like. From the calculated results, we found that clusters with even hydrogen atoms existed larger energy gaps than that with odd hydrogen atoms, which indicated that the clusters with even hydrogen atoms had higher chemical stability than that with odd hydrogen atoms. And the result was consistent with its dissociation energy. Highlights Selected hydrogen absorption of Ge3–4H n (n = 0–10) isomers and their Ge3–4H n – anions were theoretically investigated by DFT. H -bridged structure and double H-bridged structures were found in the Ge3–4H n clusters in the range of n = 1–6. The geometrical structure, relative stability, electronic properties and dissociation behaviour of Ge3–4H n (n = 0–10) cluster and their Ge3–4H n – anions were systematically analysed by a series of methods. The infrared spectra of the Ge3–4H n (n = 0–10) clusters was also simulated for further stimulated study.

Pseudo S-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε>0, the pseudo S-spectrum of T is defined asΛεS(T):=σS(T)⋃{q∈H∖σS(T):‖Qq(T)−1‖≥1ε}, where H denotes the division ring of quaternions, σS(T) is the S-spectrum of T and Qq(T)=T2−2Re(q)T+|q|2I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S-spectrum and explicitly compute the pseudo S-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε>0 the pseudo S-spectrum of the operator is the circularization of a closed disk in the complex plane centred at r with the radius ε. Further, we propose a G1-condition for quaternionic operators and prove some results in this setting.

On the Lie Centralizers of Quaternion Rings

In this paper, we investigate the problem of describing the form of Lie centralizers on quaternion rings. We provide some conditions under which a Lie centralizer on a quaternion ring is the sum of a centralizer and a center valued map.

On the structure of quaternion rings

Let S be a ring with identity in which 2 is invertible. In this paper we describe the structure of the quaternion ring R = H(S) which is a generalization of the Hamilton?s division ring of real quaternions H = H(R).

Sums of quaternion squares and a theorem of Watson

We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can be written as the sum of 3 squares.

Quaternion rings over local rings

In 1843, Hamilton (1805–1865) discovered the 4-dimensional division algebra [Formula: see text] over the field [Formula: see text] of real numbers. Hamilton’s big discovery is the following beautiful multiplications for the basis [Formula: see text] :[Formula: see text] [Formula: see text] contains the field [Formula: see text] of complex numbers; therefore [Formula: see text]. Frobenius (1849–1917) showed the following outstanding theorem. Theorem (Frobenius). Up to isomorphism, the only finite-dimensional non-commutative division algebra over [Formula: see text] is [Formula: see text]. Starting from given any ring [Formula: see text] and a free right [Formula: see text]-module [Formula: see text], the quaternion ring [Formula: see text] is canonically defined by the multiplications [Formula: see text]. For a commutative field [Formula: see text] with [Formula: see text], [Formula: see text] is a division ring or isomorphic to the ring of [Formula: see text] matrices over [Formula: see text]. This is a classical theorem. For nonzero [Formula: see text] in the center of a ring [Formula: see text], the generalized quaternion ring [Formula: see text] is defined. [Formula: see text] is [Formula: see text]. In [I. Kikumasa, K. Koike and K. Oshiro, Complex rings and quaternion rings, East-West J. Math. 21 (2019) 1–19; I. Kikumasa, G. Lee and K. Oshiro, Complex Rings, Quaternion Rings and Octonion Rings (Lambert Academic Publishing, 2020); G. Lee and K. Oshiro, Quaternion rings and octonion rings, Front. Math. China 12(1) (2017) 143–155], quaternion rings and generalized quaternion rings over division rings or other rings are studied. Now, in this paper, from ring theoretic viewpoints, we study quaternion rings [Formula: see text] and generalized quaternion rings [Formula: see text] over local rings [Formula: see text]. From our results, we can clearly look over several classical results on [Formula: see text] and [Formula: see text] over commutative fields [Formula: see text].

Factorization of Dual Quaternion Polynomials Without Study\u2019s Condition

In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of “vertical Darboux joints”. They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

Generalized Derivations and Generalized Jordan Derivations of Quaternion Rings

Generalized Derivations and Generalized Jordan Derivations of Quaternion Rings

A quaternionic Nullstellensatz

We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the quaternions, and more generally, over any division algebra .

On derivations, biderivations and superbiderivations of quaternion rings

Let $${\mathcal {S}}$$ be a ring with identity in which 2 is invertible. In this paper, we characterize derivations and biderivations of the quaternion ring $$R=H({\mathcal {S}})$$ over $${\mathcal {S}}$$. Moreover, considering R as a $${\mathbb {Z}}_2$$-graded ring, we characterize the superbidrivations of R.

Cayley–Dickson process and centrally essential rings

We describe associative center [Formula: see text] and the center [Formula: see text] of the ring [Formula: see text] obtained by applying the generalized Cayley–Dickson construction and we find conditions under which the ring [Formula: see text] is [Formula: see text]-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a nonassociative centrally essential ring.

Isomorphism From Quaternion To Matrix Ring Over Field ℤp

The quaternion ring and matrixs ring 2 by 2 over ℤp are non-commutative rings. It seams, the two rings are not identical. However, the result of the study show that the two rings area identical or isomorphic. This paper, presents several forms of isomorphism between the two rings.

Cayley–Dickson process and centrally essential rings

We describe associative center [Formula: see text] and the center [Formula: see text] of the ring [Formula: see text] obtained by applying the generalized Cayley–Dickson construction and we find conditions under which the ring [Formula: see text] is [Formula: see text]-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a nonassociative centrally essential ring.

A note on convexity of sections of quaternionic numerical range

The quaternionic numerical range of matrices over the ring of quaternions is not necessarily convex. We prove Toeplitz-Hausdorff like theorem: for any given quaternionic matrix, every section of its quaternionic numerical range is convex. We provide some additional equivalent conditions for the quaternionic numerical range of matrices over quaternions to be convex and prove some numerical radius inequalities.

On the Lie derivations and generalized Lie derivations of quaternion rings

Let S be a unital ring in which 2 is invertible, and let be the quaternion ring over S. In this paper, we describe the Lie derivations and generalized Lie derivations of R, we show that if S is commutative or semiprime, then every Lie derivation (resp. generalized Lie derivation) of R decomposes into the sum of a derivation (resp. generalized derivation) and an additive central map that vanishes on all commutators of R.

Generalized Quaternion Rings over $$\mathbb {Z}/n\mathbb {Z}$$ Z / n Z for an Odd $$\varvec{n}$$ n

We consider a generalization of the quaternion ring \(\Big (\frac{a,b}{R}\Big )\) over a commutative unital ring R that includes the case when a and b are not units of R. In this paper, we focus on the case \(R=\mathbb {Z}/n\mathbb {Z}\) for and odd n. In particular, for every odd integer n we compute the number of non R-isomorphic generalized quaternion rings \(\Big (\frac{a,b}{\mathbb {Z}/n\mathbb {Z}}\Big )\).