Quaternion Rings Research Articles

On the idempotents, nilpotents, units and zero-divisors of finite rings

In this article, first we find the number of idempotents and the zero-divisors of a matrix ring over a finite field F. Next, given the order of the Jacobson radical of the finite unital ring R, we find the number of units, nilpotents and zero-divisors of R. Moreover, we find an upper bound for the number of idempotents of a finite ring which is in general smaller than the upper bound found by MacHale [Proc. R. Ir. 1982;82A(1):9–12]. Finally, we find the above-mentioned numbers in some matrix rings and quaternion rings.

Some questions concerning superderivations on $${\mathbb {Z}}_{2}$$ Z 2 -graded rings

In this paper we pose some questions about superderivations on $${\mathbb {Z}}_{2}$$ -graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special $${\mathbb {Z}}_{2}$$ -gradings and we check the answer to these questions about them.

Sums of squares in quaternion rings

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many Quaternion rings, and give global upper and lower bounds.

Armendariz group rings

ABSTRACTFor a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.

Quaternion rings and octonion rings

In this paper, for rings R, we introduce complex rings ℂ(R), quaternion rings ℍ(R), and octonion rings O(R), which are extension rings of R; R ⊂ ℂ(R) ⊂ ℍ(R) ⊂ O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ℂ(R) and ℍ(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.

Sum Range of a Quaternion Series

In this paper, we obtain a result which implies, in particular, that for a quaternion z ∉ {−1, 1} with |z| = 1, the sum range of the series \( {\displaystyle \sum_n\frac{z^n}{n}} \) is a closed proper subfield of the division ring of quaternions ℍ isometrically isomorphic to the field of complex numbers ℂ.

Idempotents in the group ring of quaternions

In this paper we give the explicit expressions of idempotents in the group ring of quaternions over finite field Zp.

Non-commutative digit expansions for arithmetic on supersingular elliptic curves

We study non-commutative digit expansions in quaternion rings that arise in the context of supersingular elliptic curves. These digit expansions can be used in a \({\tau}\)-and-add method to speed up arithmetic (scalar multiplication and pairing) on certain families of supersingular elliptic curves in characteristic \({p \geqq 5}\). The basis \({\tau}\) is a quadratic algebraic integer that represents the Frobenius endomorphism of the curve, which is a very fast operation to evaluate. We prove the existence of a finite expansion for every element of the quaternion ring, as well as the equivalence between right and left digit expansions (i.e. the basis \({\tau}\) is placed right, resp. left, to the digit); this expansion turns out to be a non-adjacent form (NAF) for integers, i.e. in every two consecutive digits there is at least one 0.

Directed partial orders on real quaternions

It is shown that a division ring of real quaternions can be made into a partially ordered ring with a directed partial order.

Cryptanalysis of the Quaternion Rainbow

Rainbow is one of signature schemes based on the problem solving a set of multivariate quadratic equations. While its signature generation and verification are fast and the security is presently sufficient under suitable parameter selections, the key size is relatively large. Recently, Quaternion Rainbow — Rainbow over a quaternion ring — was proposed by Yasuda, Sakurai and Takagi (CT-RSA'12) to reduce the key size of Rainbow without impairing the security. However, a new vulnerability emerges from the structure of quaternion ring; in fact, Thomae (SCN'12) found that Quaternion Rainbow is less secure than the same-size original Rainbow. In the present paper, we further study the structure of Quaternion Rainbow and show that Quaternion Rainbow is one of sparse versions of the Rainbow. Its sparse structure causes a vulnerability of Quaternion Rainbow. Especially, we find that Quaternion Rainbow over even characteristic field, whose security level is estimated as about the original Rainbow of at most 3/4 by Thomae's analysis, is almost as secure as the original Rainbow of at most 1/4-size.

To Principles of Quantum Theory Construction

New insight to the principles of the quantum theory construction is given. It is based on the symmetry study of main differential equations of mechanics and electrodynamics. It has been shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered within the frames of the number theory) determine the mathematical nature of the quantities, incoming in given equations. The substantial consequence of the given consideration is the proof of the main postulate of quantum mechanics, that is, the proof of the statement, that to any quantum mechanical quantity can be set up into the correspondence the Hermitian matrix. It is shown, that a non-abelian character of the multiplicative group of the quaternion ring leads to the nonapplicability of the quaternion calculus for the construction of new versions of quantum mechanics directly. The given conclusion seems to be actual, since there is a number of modern publications with the development of the quantum mechanics theory using the quaternions with the standard basis {e, i, j, k}. The correct way for the construction of new versions of quantum mechanics on the quaternion base is discussed in the paper presented. It is realized by means of the representation of the quaternions through the basis of the linear space of complex numbers over the field of real numbers, under the multiplicative group of which the equations of the dynamics of mechanical systems are invariant. At the same time the quaternion calculus is applicable in electrodynamics, at that the new versions of quantum electrodynamics can be constructed by an infinite number of the ways corresponding to an infinite number of the matrix representations of the standard quaternion basis {e, i, j, k}. The given conclusion is the consequence of the high symmetry of Maxwell equations.

A Proxy Signature Based on the Difficulty of Solving Equations of Higher on Quaternion Ring

At present, all the applied schemes of the digital signature schemes based on the research of the factoring problem, discrete logarithm problem, and discrete logarithm problem on the elliptic curve cant meet the demand of actual application. This paper introduces the knowledge to the digital signature and studies the signature scheme based on the quaternion ring. The scheme introduces the multivariate functions of high degree to non-commutative quaternion ring, and its safety based on the computational difficulty to solve the the multivariate equations of high degree. The article applies this thought to proxy signature and designs a proxy signature scheme. Finally, it gives the safety analysis of the new scheme.

Signature Scheme Using the Root Extraction Problem on Quaternions

The root extraction problem over quaternion rings modulo an RSA integer is defined, and the intractability of the problem is examined. A signature scheme is constructed based on the root extraction problem. It is proven that an adversary can forge a signature on a message if and only if he can extract the roots for some quaternion integers. The performance and other security related issues are also discussed.

On addition and multiplication of points in a certain class of projective Klingenberg planes

Let ( O , E , U , V ) Open image in new window be the coordination quadruple of the projective Klingenberg plane (PK-plane) coordinated with dual quaternion ring Q ( e ) = Q + Q e = { x + y e ∣ x , y ∈ Q } Open image in new window, where Q is any quaternion ring over a field. In this paper, we define addition and multiplication of points on the line O U = [ 0 , 1 , 0 ] Open image in new window geometrically, also we give the algebraic correspondences of them. Finally, we carry over some well-known properties of ordinary addition and multiplication to our definition.

The collineations which act as addition and multiplication on points in a certain class of projective Klingenberg planes

Let P K 2 (Q(ε)) be the projective Klingenberg plane coordinated by the dual quaternion ring Q(ε)=Q+Qε={x+yε∣x,y∈Q} where Q is any quaternion ring. In this paper, we determine the addition and multiplication of the points on the line [0,1,0] of P K 2 (Q(ε)) as the image of some collineations of the plane P K 2 (Q(ε)). To do this, we give the collineations S a and L a . Later we show that the addition and multiplication of the nonneighbor points on the line [0,1,0] can be obtained as the images under that S a and L a .MSC:51C05, 51J10, 12E15.

When is the ring of 2x2 matrices over a ring Galois?

Let R be a ring with 1, M2(R) the ring of 2 × 2 matrices over R, and S the Sugano quaternion ring over R. Then, S is isomorphic with a subring of M2(R) if and only if 2 is not a zero divisor in R, and S ∼ M2(R) if and only if 2 is invertible in R. Moreover, if 2 is invertible in R, then M2(R) is a Galois extension of R with an abelian inner Galois group G of order 4. This implies that the rings of 2 × 2 matrices over the real field and complex field are central Galois algebras induced by the central Galois algebra of 2 × 2 matrices over the rational field with Galois group G. Mathematics Subject Classification: 16S35, 16W20

Integer-Valued Polynomials on the Hurwitz Ring of Integral Quaternions

Let ℍ denote the ring of Hurwitz quaternions, the maximal order in the ring of rational quaternions, ℍ(ℚ), and let Int(ℍ) = {f(x) ∈ ℍ(ℚ)[x]: f(ℍ) ⊆ ℍ} be the ℍ-algebra of quaternionic polynomials taking values in ℍ at elements of ℍ. Also let c n denote a generator for the left ideal of elements c of ℍ with the property that c·f(x) ∈ ℍ[x] for all elements f ∈ Int(ℍ) of degree ≤n. In this article, we give upper and lower bounds for the power of the prime 1 + i dividing c n and show that these coincide for n ≤ 6.

Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

Abstract It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $\mathbb{F}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where $\mathbb{F}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, $\mathbb{F}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.

Characterizing quaternion rings over an arbitrary base

We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra.

Quantum–mechanical and quantum–electrodynamic equations for spectroscopic transitions

Transition operator method is developed for description of the dynamics of spectroscopic transitions. Quantum–mechanical and quantum–electrodynamic difference-differential equations in general discrete space case and differential equations in continuum limit have been derived for spectroscopic transitions in the system of periodical ferroelectrically (ferromagnetically) ordered chains, interacting with external electromagnetic field. It was shown, that given equations can be represented in the form of Landau–Lifshitz equation in continuum limit and its generalization in discrete space case. Landau–Lifshitz equation was represented in Lorentz invariant form by Hilbert space definition over the ring of quaternions. It has been shown, that spin vector can be considered to be quaternion vector of the state of the system studied. From comparison with pure optical experiments the value of spin S=1/2 for spin-Peierls solitons in carbon chains has been found and it has also been established, that given quasiparticles are dually charged. The ratio of magnetic to electric (imaginary to real) components of electromagnetic dual (complex) charge is evaluated for given centers to be g e ≈ ( 1.1 − 1.3 ) 10 2 in correspondence with Dirac theory of charge quantization. The given results seem to be obtained for the first time.