Ideals In Rings Research Articles

MacWilliams Identities and Generator Matrices for Linear Codes over ℤp4[u]/(u2 − p3β, pu)

Suppose that R=Zp4[u] with u2=p3β and pu=0, where p is a prime and β is a unit in R. Then, R is a local non-chain ring of order p5 with a unique maximal ideal J=(p,u) and a residue field of order p. A linear code C of length N over R is an R-submodule of RN. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p≠2, there are precisely two distinct rings with these properties up to isomorphism. However, for p=2, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R, illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings.

Combinatorial Optimization Systems Theory Prospected from Rotational Symmetry

Combinatorial optimization systems theory prospected from rotational symmetry involves techniques for improving the quality indices of engineering devices or systems with non-uniform structure (e.g., controllable cyber-physical objects) concerning transformation swiftness, position accuracy, and resolution, using designs based on extraordinary geometric properties and structural excellence of combinatorial conformations, namely the concept of Ideal Ring Bundles. Design techniques based on the underlying combinatorial theory provide configure one- and multidimensional systems with smaller amounts of elements than at present, while maintaining the other substantial operating characteristics of the systems.

Noise Level of a Ring Laser Gyroscope in the Femto-Rad/s Range.

Absolute angular rotation rate measurements with sensitivity better than prad/s would be beneficial for fundamental science investigations. In this regard, large frame Earth based ring laser gyroscopes are top instrumentation as far as bandwidth, long-term operation, and sensitivity are concerned. Here, we demonstrate that the GINGERINO active-ring laser upper limiting noise is close to 2×10^{-15} rad/s for ∼2×10^{5} s of integration time, as estimated by the Allan deviation evaluated in a differential measurement scheme. This result is more than a factor of 10 better than the theoretical prediction so far accounted for ideal ring lasers shot-noise with the two beams counterpropagating inside the cavity considered as two independent propagating modes. This feature is related to the peculiarity of real ring laser system dynamics that causes phase crosstalking among the two counterpropagating modes. In this context, the independent beam model is, then, not applicable, and the measured noise limit falls below the expected one.

Combinatorial Optimization of Engineering Systems based on Diagrammatic Design

The objectives of the combinatorial optimization of engineering systems based on diagrammatic design are enhancing technical indices of the systems with spatially or temporally distributed elements (e.g., radio-antenna arrays) concerning resolving ability, positioning precision, transmission speed, and performance reliability, using the graphical performance of appropriate algebraic models of the system, such as cyclic difference sets, Galois fields and “Ideal Ring Bundles”. The diagrammatic design provides configuring systems with a smaller number of elements than at present, while upholding or improving on the other significant operating quality indices of the system.

Perfect Zeeman Anisotropy in Rotationally Symmetric Quantum Dots with Strong Spin-Orbit Interaction.

In nanoscale structures with rotational symmetry, such as quantum rings, the orbital motion of electrons combined with a spin-orbit interaction can produce a very strong and anisotropic Zeeman effect. Since symmetry is sensitive to electric fields, ring-like geometries provide an opportunity to manipulate magnetic properties over an exceptionally wide range. In this work, we show that it is possible to form rotationally symmetric confinement potentials inside a semiconductor quantum dot, resulting in electron orbitals with large orbital angular momentum and strong spin-orbit interactions. We find complete suppression of Zeeman spin splitting for magnetic fields applied in the quantum dot plane, similar to the expected behavior of an ideal quantum ring. Spin splitting reappears as orbital interactions are activated with symmetry-breaking electric fields. For two valence electrons, representing a common basis for spin-qubits, we find that modulating the rotational symmetry may offer new prospects for realizing tunable protection and interaction of spin-orbital states.

Binomial ideals attached to finite collections of cells

We consider the ideal of inner 2-minors I P of a finite set of cells P , which we call the cell ideal of P . A nice interpretation for the height of an unmixed ideal I P , in terms of the number of cells of P is given. Moreover, the coordinate rings of cell ideals with isolated singularities are determined.

When do modules mimic arbitrary sets?

We study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism A → B , there is a monomorphism B → A (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, we prove that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky’s dual rings) is right Artinian and right serial. Consequently, R is an Artinian PIR if and only if every factor ring of R cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. Some known results and the above one yield the following: A ring R is a direct sum of right Artinian right chain rings and Artinian PIR’s if and only if every factor ring of R cogenerates its (uniform) cyclic right modules (with nonzero socle); so, such rings coincide with the right CES-rings of Jain and Lopez-Pérmouth, rings whose factors are right CF and rings that satisfy the above mentioned property for their cyclic right modules A and B. Finally, a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other.

Prime Principal Right Ideal Rings

Prime Principal Right Ideal Rings

Regularity of the edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.

Solving systems of algebraic equations over finite commutative rings and applications

AbstractSeveral problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

Ideal factoriality, semistar operations, and quasiprincipal Ideals

The w-operation, a “smoother” variant of the classic t-operation on the set of ideals of a commutative ring R, has recently attracted significant attention. We study when R’s monoid of w-ideals is “factorial” in some sense. For example, we show that every proper w-ideal of R has a unique up to order irredundant w-factorization into w-unfactorable w-ideals if and only if R is a finite direct product of Krull domains and principal ideal rings, if and only if every w-ideal is w-quasiprincipal. We prove several analogous results regarding other kinds of “w-ideal factoriality” and generalize these results via semistar operations.

On Linear Codes over Finite Singleton Local Rings

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z4 codes, which play a significant role in coding theory.

Combinatorial Optimization Systems Theory Prospected from Rotational Symmetry

Combinatorial optimization systems theory prospected from rotational symmetry involves techniques for improving the quality indices of engineering devices or systems with non-uniform structure (e.g., controllable cyber-physical objects) concerning transformation swiftness, position accuracy, and resolution, using designs based on extraordinary geometric properties and structural excellence of combinatorial conformations, namely the concept of Ideal Ring Bundles. Design techniques based on the underlying combinatorial theory provide configure one- and multidimensional systems with smaller amount elements than at present, while maintaining the other substantial operating characteristics of the systems.

Spectral equivalence of smooth group schemes over principal ideal local rings

Let G be a smooth linear group scheme of finite type. For any positive integer k and a finite field F, let Wk(F) be the ring of Witt vectors of length k over F. We show that the group algebras of G(F[t]/(tk)) and G(Wk(F)) are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer k and finite field F with large enough characteristic. We also prove that if charF is large enough, then the cardinality of the set {dim⁡ρ|ρ∈irr(G(F))} is bounded uniformly in F.

Rees algebras of unit interval determinantal facet ideals

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebras and special fiber rings of unit interval determinantal facet ideals. In particular, we show that unit interval determinantal facet ideals are of fiber type and that their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.

On the rank decoding problem over finite principal ideal rings

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings.In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

IDEALS AND ALMOST PRINCIPAL IDEALS IN EUCLIDEAN Γ−SEMIRINGS

The generalization of the ring of ordinary integers and their properties into an Euclidean ring is well known. Every ideal in an Euclidean ring is a principal ideal, as is also widely known. That is, the Euclidean ring is a principal ideal ring. This paper aims to generalize the Γ−semiring of non-negative integers and their properties by defining Euclidean Γ−semiring. A Euclidean Γ−semiring is one of the many special classes of Γ−semirings. Finally, the special class of Γ−semirings discussed in this paper is the class of almost principal ideal Γ−semirings.

Novel Magnetic Field Modeling Method for a Low-Speed, High-Torque External-Rotor Permanent-Magnet Synchronous Motor

In view of the unstable electromagnetic performance of the air gap magnetic field caused by the torque ripple and harmonic interference of a multi-slot and multi-pole low-speed, high-torque permanent magnet synchronous motor, we propose a simplified model of double-layer permanent magnets. The model is divided into an upper and a lower subdomain, with the upper subdomain being an ideal circular ring and the lower subdomain being a segmented sector ring. Moreover, we develop an exact analytical model of the motor that predicts the magnetic field distribution based on Laplace’s and Poisson’s equations, which is solved using the method of separating variables. Taking a 40p168s low-speed, high-torque permanent magnet synchronous motor as an example, the accuracy of the model is verified by comparison with an ideal circular ring model, a segmented sector ring model, and the finite element method. Based on the proposed simplified model, three combined permanent magnets considering both edge-cutting and polar arc cutting structures are proposed, which are chamfered, rounded, and rectangular combinations. Under the premise of a consistent edge-cutting amount, the electromagnetic characteristics of the three combination types of permanent magnets are compared using the finite element method. The results show that the electromagnetic characteristics of the chamfered combination PM are superior to those of the other two combinations. Finally, a prototype is manufactured and tested to validate the theoretical analysis.

Depth and Stanley Depth of the Edge Ideals of r-Fold Bristled Graphs of Some Graphs

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs.

Optimum Vector Information Technologies Based on the Multidimensional Combinatorial Configurations

Paper devoiced to optimum vector information technologies based on the multi-dimensional combinatorial configurations, such as Ideal Ring Bundles (IRBs). One-dimensional IRBs are ring ordered positive integers that form finite set of integers from 1 to S using both these numbers and all its consecutive terms. Two- and multi-dimensional IRBs make available to configure intelligent information and telecommunication systems providing generate the maximum number of distinct vector sums of consecutive terms in the combinatorial configuration. Applications profiting from optimum vector information technologies based on the multi-dimensional combinatorial theory provide for example data mining technologies and big vector data processing, data analysis and system security, signal compression and reconstruction, vector computing and telecommunications, and other branches of sciences and advanced information technologies.