Noncommutative Rings Research Articles

The Witt vectors for Green functors

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology THHCn(−), and under flatness conditions it describes the E2 term of the Künneth spectral sequence for relative THH. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.

A Characterization of Derivations in Prime Rings with Involution

The purpose of this paper is to investigate $*$-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring $R$ admit nonzero derivations $d_1, d_2$ such that $[d_1(x), d_2(x^*)]=0$ for all $x\in R$, then $d_1=\lambda d_2$, where $\lambda\in C$. Finally, we provide an example to show that the condition imposed in the hypothesis of our results are necessary.

On group modules

The paper is focused on questions when some homological and submodule-chain conditions satisfied by a module [Formula: see text] are preserved by the group module [Formula: see text]. Namely, it is proved for a group [Formula: see text] and an [Formula: see text]-module [Formula: see text] that [Formula: see text] is flat if and only if [Formula: see text] is flat, and [Formula: see text] is artinian if and only if [Formula: see text] is artinian and [Formula: see text] is finite, which are two questions raised by Yiqiang Zhou: On Modules Over Group Rings, Noncommutative Rings and Their Applications LENS July 1-4, 2013.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and fixed positive integers. In the present paper, we describe the structure of R and all possible forms of G and g in the following cases: for all and for all

Almost prime and weakly prime submodules

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a right [Formula: see text]-module. A proper submodule [Formula: see text] of [Formula: see text] is called almost prime (respectively, weakly prime) if for each submodule [Formula: see text] of [Formula: see text] and each ideal [Formula: see text] of [Formula: see text] that [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] or [Formula: see text]. We study these notions which are new generalizations of the prime submodules over noncommutative rings and we obtain some related results. We show that these two concepts in some classes of modules coincide. Moreover, we investigate the conditions that [Formula: see text] is almost prime, where [Formula: see text] is a submodule of [Formula: see text] and [Formula: see text] is an ideal of [Formula: see text]. Also, the almost prime radical of modules will be introduced and we extend some known results.

On diameter of the zero-divisor and the compressed zero-divisor graphs of skew Laurent polynomial rings

Let [Formula: see text] be an associative ring with nonzero identity. The zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonzero zero-divisors of [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the set of all right annihilators and the set of all left annihilator of an element [Formula: see text], respectively, and let [Formula: see text]. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of [Formula: see text] and [Formula: see text], where the base ring [Formula: see text] is reversible and also has the [Formula: see text]-compatible property.

Quaternionic representation of electromagnetism for material media

In this study, the electromagnetic fields are developed in the presence of both the electric and magnetic induction fields by quaternion algebra. In this sense, the polarization and magnetization effects, which are valid in the material media, gain much importance. Quaternions are one of the most convenient tools for representing electromagnetism with regard to having non-commutative but associative algebraic division ring. By defining the quaternion induction field, the quaternion source term has been obtained in basic and elegant notation for the first time. In addition, one type of Poynting theorem, named as the Minkowski form, has been presented including the permittivity and permeability constants by quaternions.

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.

Maximal subgroups of SLn(D)

Given a non-commutative F-central division ring D, N a subnormal subgroup of GLn(D) and M a non-abelian maximal subgroup of N, if M is a non-abelian soluble maximal subgroup of N, then, n=1 and D is cyclic of prime degree p with a maximal cyclic subfield K/F such that the groups Gal(K/F) and M/(K⁎∩M) are isomorphic. Furthermore, for any x∈M∖K⁎, we have xp∈F⁎ and D=F[M]=⨁i=1pKxi.

Artinian bimodule with quasi-Frobenius bimodule of translations

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

New Forms of Defining the Hidden Discrete Logarithm Problem

There are introduced novel variants of defining the discrete logarithm problem in a hidden group, which represents interest for constructing post-quantum cryptographic protocols and algorithms. This problem is formulated over finite associative algebras with non-commutative multiplication operation. In the known variant this problem, called congruent logarithm, is formulated as superposition of exponentiation operation and automorphic mapping of the algebra that is a finite non-commutative ring. Earlier it has been shown that congruent logarithm problem defined in the finite quaternion algebra can be reduced to discrete logarithm in the finite field that is an extension of the field over which the quaternion algebra is defined. Therefore further investigations of the congruent logarithm problem as primitive of the post-quantum cryptoschemes should be carried out in direction of finding new its carriers. The present paper introduces novel associative algebras possessing significantly different properties than quaternion algebra, in particular they contain no global unit. This difference had demanded a new definition of the discrete logarithm problem in a hidden group, which is different from the congruent logarithm. There are proposed several variants of such definition, in which it is used the notion of the local unite. There are considered right, left, and bi-side local unites. Two general methods for constructing the finite associative algebras with non-commutative multiplication operation are proposed. The first method relates to defining the algebras having dimension value equal to a natural number m > 1, and the second one relates to defining the algebras having arbitrary even dimensions. For the first time the digital signature algorithms based on computational difficulty of the discrete logarithm problem in a hidden group have been proposed.

Lie centrally metabelian group rings over noncommutative rings

Let R be a unital ring and let G be an arbitrary nonabelian group. Lie centrally metabelian group algebras have been studied and described by various authors. In this article, we have classified Lie centrally metabelian group rings over noncommutative rings. We also give characterizations of group rings RG satisfying the Lie identity over both commutative and noncommutative rings.

A note on super integral rings

Let $R$ be a nite non-commutative ring with center $Z(R)$. The commuting graph of $R$, denoted by $\Gamma_R$, is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx$. Let$\Spec(\Gamma_R), \L-Spec(\GammaR)$ and $\Q-Spec(\GammaR)$ denote the spectrum, Laplacian spectrum and signless Laplacian spectrum of $\Gamma_R$ respectively. A nite non-commutative ring $R$ is called super integral if $\Spec(\Gamma_R), \L-Spec(Gamma_R)$ and $\Q-Spec(\Gamma_R)$ contain only integers. In this paper, we obtain several classes of super integral rings.

Some new linear codes from skew cyclic codes and computer algebra challenges

One of the main problems of coding theory is to construct codes with best possible parameters. Cyclic codes and their various generalizations, such as quasi-twisted codes, have been a fruitful source in achieving this goal. Recently, a new generalization of cyclic codes that are known as skew cyclic codes have been introduced and some new codes obtained from this class. Unlike many other types of codes considered in algebraic coding theory, skew cyclic codes require one to work in a non-commutative ring called skew polynomial ring. In this paper, we present some new linear codes obtained from the class of skew cyclic codes and describe computational challenges in working with this class of codes.

Generalized skew-derivations and generalization of homomorphism maps in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, Qr the right Martindale quotient ring of R, the extended centroid of R and a noncentral multilinear polynomial over C. In this paper we describe all possible forms of two generalized skew derivations F and G of R satisfying the condition , for all .

On clean and regular elements of noncommutative ring extensions

Let R be an associative ring with identity and α be an endomorphism of R. In this article, we are interested to study the some of relations between a ring R and that of D. A. Jordan’s construction of the ring A(R,α) as well as the skew Laurent polynomial ring . The main propose of this article is to characterize the unit elements, the idempotent elements, von Neumann regular elements, π-regular elements, von Neumann local elements and also the clean elements of the skew Laurent polynomial ring as well as Jordan’s construction of the ring A(R,α). Applying these characterizations, one can easily get some nice radical-theoretic properties of the mentioned classes of rings.

Skew Generalized Power Series Rings and the McCoy Property

Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.

ON 2-ABSORBING MODULES OVER NONCOMMUTATIVE RINGS

Let R be a noncommutative ring with identity. We de ne the notion of a 2-absorbing submodule and show that if the ring is commutative then the notion is the same as the original de nition of that of A. Darani and F. Soheilnia. We give an example to show that in general these two notions are di erent. Many properties of 2-absorbing submodules are proved which are similar to the results for commutative rings.