Quotient Ring Research Articles

Generalized derivations preserving Engel condition in prime and semiprime rings

Let [Formula: see text] be a prime ring with [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] its Martindale quotient ring and [Formula: see text] its extended centroid. Let [Formula: see text] and [Formula: see text] be nonzero generalized derivations on [Formula: see text] such that [Formula: see text] Then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text], for any [Formula: see text], unless [Formula: see text], where [Formula: see text] is the algebraic closure of [Formula: see text].

Power Values of Generalized Skew Derivations with Annihilator Conditions on Lie Ideals

Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations $$\delta $$ and $$\varrho $$ of R in case of $$\begin{aligned} p(\delta (u)^{l_{1}}\varrho (u)^{l_{2}}\delta (u)^{l_{3}}\varrho (u)^{l_{4}}\cdots \varrho (u)^{l_{k}} )^{n}=0, \end{aligned}$$ for all $$u\in L$$ , where $$0 \ne p\in R$$ , $$l_{1},l_{2},\ldots ,l_{k}$$ are fixed nonnegative integers with $$l_{1}\ne 0$$ and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if $$p\delta (u)^{n}=0$$ for all $$u \in L$$ , then there exists $$a\in Q$$ such that $$\delta (x)=ax$$ for all $$x \in R$$ and $$pa=0$$ unless R satisfies $$s_4$$ .

On free subgroups in division rings

Let K K be a field, let σ \sigma be an automorphism, and let δ \delta be a σ \sigma -derivation of K K . We show that the multiplicative group of nonzero elements of the division ring D = K ( x ; σ , δ ) D=K(x;\sigma ,\delta ) contains a free noncyclic subgroup unless D D is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free nonabelian solvable-by-finite groups always contain free noncyclic subgroups.

New Orthogonal Binary Sequences Using Quotient Rings Z/<i>n</i>Z Where <i>n</i> Is a Multiple of Some Prime Numbers

Orthogonal Sequences (as M-Sequences, Walsh Sequences,…) are used widely at the forward links of communication channels to mix the information on connecting to and at the backward links of these channels to sift through this information is transmitted to reach the receivers this information in a correct form, especially in the pilot channels, the Sync channels, and the Traffic channel. This research is useful to generate new sets of orthogonal sequences (with the bigger lengths and the bigger minimum distance that assists to increase secrecy of these information and increase the possibility of correcting mistakes resulting in the channels of communication) from quotient rings Z/nZ, where Z is the integers and n is not of the form pm, where p is prime, replacing each event number by zero and each odd number by one, also, the increase in the natural number does not necessarily lead to an increase in the size of the biggest orthogonal set in the corresponding quotient ring. The length of any sequence in a biggest orthogonal set in the quotient ring Z/nZ is n and the minimum distance is between (n-3)/2 and (n-1)/2 and the sequences can be used as keywords or passwords for secret messages.

On constructing boundaries for boundary value problems defined by continuous 2-D autonomous systems

Abstract In this paper, we provide a constructive way of specifying initial/boundary data for a given continuous 2-D autonomous system described by a set of linear partial differential equations (PDEs) with real constant coefficients. One of the ways of specifying initial/boundary data is by specifying the values of various derivatives of the solution trajectories at the origin; the derivatives correspond to a standard monomial set obtained using Grobner basis. However, such an initial/boundary data often lacks physical interpretation. In this paper, we consider subsets of the domain having some algebraic structure (in the form of subspaces and strips of finite width around such subspaces) such that trajectories restricted to these subsets, often called characteristic sets, serve as initial/boundary conditions for the given autonomous system of linear PDEs. We provide a systematic way to construct such characteristic sets with the help of Grobner bases and Oberst-Riquier algorithm. Thus we bridge the gap between initial/boundary conditions involving standard monomials and more conventional initial/boundary conditions in the form of restrictions on characteristic sets. We also show that every scalar system of PDEs admits such a characteristic set given by a rectangular strip of finite width around a subspace whose dimension equals the Krull dimension of the system’s quotient ring.

On a generalization of self-injective rings

In this work the notion of left (right) self-injective ring is generalized. We consider rings that are direct sum of injective module and semisimple module as a left (respectively, right) module above itself. We call such rings left (right) semi-injective and research their properties with the help of two-sided Peirce decomposition of the ring. The paper contains the description of left noetherian left semi-injective rings. It is proved that any such ring is a direct product of (two-sided) self-injective ring and several quotient rings (of special kind) of rings of upper triangular matrices over skew fields. From this description it follows that for left semi-injective rings we have the analogue of the classical result for self-injective rings. Namely, if a ring is left noetherian and left semi-injective then this ring is also right semi-injective and two-sided artinian.

Construction of Signal Sets From Quotient Rings of the Quaternion Orders Associated With Arithmetic Fuchsian Groups

This paper aims to construct signal sets from quotient rings of the quaternion over a real number field associated with the arithmetic Fuchsian group Γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4g</sub> , where g is the genus of the associated surface. These Fuchsian groups consist of the edge-pairing isometries of the regular hyperbolic polygons (fundamental region) P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4g</sub> , which tessellate the hyperbolic plane D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The corresponding tessellations are the self-dual tessellations {4g, 4g}. Knowing the generators of the quaternion orders which realize the edge-pairings of the polygons, the signal points of the signal sets derived from the quotient rings of the quaternion orders are determined. It is shown by examples the relevance of adequately selecting the ideal in the maximal order to construct the signal sets satisfying the property of geometrical uniformity. The labeling of such signals is realized by using the mapping by set partitioning concept to solve the corresponding Diophantine equations (extreme quadratic forms). Trellis coded modulation and multilevel codes whose signal sets are derived from quotient rings of quaternion orders are considered possible applications.

Efficient and Secure Implementation of NTRUEncrypt Using Signed Sliding Window Method

NTRUEncrypt is a public key cryptosystem based on hard problems over lattices. The dominant operation in NTRUEncrypt is convolution, i.e., multiplication over a quotient ring of polynomials. Based on the fact that a convolution has a highly regular structure, Lee et al. proposed the sliding window method for fast convolution of binary polynomials in 2013, which was then extended to ternary polynomials for ideal lattices by Akleylek, Alkim, and Tok in 2016. These sliding window methods reduce the cost of a convolution operation using look-up tables that store partial computation results related to repeated coefficient patterns. In this paper, we propose a signed sliding window method with side-channel resistance for NTRUEncrypt. The proposed method considers both positive and negative nonzero coefficients when constructing look-up tables. The new method not only accelerates convolution but also enables the application of power analysis countermeasures effectively. According to the experimental results, the constant-time implementation of the proposed method with timing and power analysis countermeasures accelerates the previously developed secure convolution method by up to 20%.

A note on generalized derivations with multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R with the extended centroid C and be a non-central multilinear polynomial over C. Suppose that F and G are two generalized derivations on R such that for all then we give the complete description of the mappings F and G.

English

Let R be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring Q, C the extended centroid of R and a ∈ R. Suppose that δ is a nonzero σ-derivation of R such that a[δ(xn), xn]k = 0 for all x ∈ R, where σ is an automorphism of R, n and k are fixed positive integers. Then a = 0.

Depth and Stanley depth of the edge ideals of the powers of paths and cycles

Abstract Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

Composition and orthogonality of derivations with multilinear polynomials in prime rings

Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and $$\delta $$ be two derivations of R and S be the set of evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ over C which is not central valued. Let $$p,q\in R$$ . We prove the followings.

Generalized Skew Derivations and Nilpotent Values on Lie Ideals

Let R be a prime ring of characteristic different from 2 and 3, Qr be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x))n = 0 for all x ∈ L, then one of the following holds: (a) there exists c ∈ Qr such that F(x) = xc and G(x) = cx; (b) R satisfies s4 and there exist a, b, c ∈ Qr such that F(x) = ax + xc, G(x) = cx + xb and (a − b)2 = 0.

On the radius of the category of extensions of matrix factorizations

Let S be a commutative noetherian ring. The extensions of matrix factorizations of non-zerodivisors x1,…,xn of S form a full subcategory of finitely generated modules over the quotient ring S/(x1⋯xn). In this paper, we investigate the radius (in the sense of Dao and Takahashi) of this full subcategory. As an application, we obtain an upper bound of the dimension (in the sense of Rouquier) of the singularity category of a local hypersurface of dimension one, which refines a recent result of Kawasaki, Nakamura and Shimada.

Covering classes, strongly flat modules, and completions

We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class $$ \mathcal {SF}$$ of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether $$ \mathcal {SF}$$ a covering class implies $$ \mathcal {SF}$$ closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat.

Generalized skew-derivations acting on Lie ideals with central values in prime rings

Let R be a prime ring with characteristic different from 2 and 3, $$Q_r$$ be its right Martindale quotient ring, C be its extended centroid, F a nonzero generalized skew derivation of R and L a noncentral Lie ideal of R. If $$\begin{aligned}{}[[F(u),u],F(u)]\in Z(R) \end{aligned}$$for all $$u\in L$$, then one of the following holds:

Algebraic lattices via polynomial rings

Signal constellations having lattice structure have been studied as meaningful means for signal transmission over Gaussian channel. Usually the problem of finding good signal constellations for a Gaussian channel is associated with the search for lattices with high packing density, where in general the packing density is usually hard to estimate. The aim of this paper was to illustrate the fact that the polynomial ring $$\mathbb {Z}[x]$$ can produce lattices with maximum achievable center density, where $$\mathbb {Z}$$ is the ring of rational integers. Essentially, the method consists of constructing a generator matrix from a quotient ring of $$\mathbb {Z}[x]$$ .

ON &lt;em&gt;m&lt;/em&gt;TH-COMMUTATORS AND ANTI-COMMUTATORS INVOLVING GENERALIZED DERIVATIONS IN PRIME RINGS

In this manuscript, we study the $m$-th commutator and anti-commutator involving generalized derivations on some suitable subsets of rings. We attain the information about the structure of rings and the behaviour of generalized derivation in form of multiplication by some specific element of Utumi quotient ring which satisfies certain differential identities.

Digraphs associated with the kth power map on the quotient ring of polynomials over finite fields

Let [Formula: see text] be a monic polynomial over a finite field [Formula: see text], and [Formula: see text] an integer. A digraph [Formula: see text] is one whose vertex set is [Formula: see text] and for which there is a directed edge from a polynomial [Formula: see text] to [Formula: see text] if [Formula: see text] in [Formula: see text]. If [Formula: see text], where the [Formula: see text]’s are distinct monic irreducible polynomials over [Formula: see text], then [Formula: see text] can be factorized as [Formula: see text]. In this work, we investigate the structure of these power digraphs. The semiregularity property is examined, and its relationship with the symmetric property is established. In addition, we look into the uniqueness of factorization of trees attached to a fixed point.

Hulls of cyclic codes over [formula omitted

The hulls of linear and cyclic codes over finite fields have been of interest and extensively studied due to their wide applications. In this paper, the hulls of cyclic codes of odd length n over the ring Z4 have been focused on. Their characterization has been established in terms of the generators viewed as ideals in the quotient ring Z4[x]∕〈xn−1〉. An algorithm for computing the types of the hulls of cyclic codes of arbitrary odd length over Z4 has been given. The 2-dimensions of the hulls of cyclic codes of length n over Z4 and the number of cyclic codes of length n over Z4 having hulls of a given 2-dimension are determined. The average 2-dimension E(n) of the hulls of cyclic codes of odd length n over Z4 has been established. A general formula for E(n) has been established together with its upper and lower bounds. It turns out that E(n) grows the same rate as n. A brief discussion on hulls of cyclic codes over Zp2, where p is an odd prime, is provided as well.