Finite Rings Research Articles

Morita equivalence for graded rings

The classical Morita Theorem for rings established the equivalence of three statements involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.

Strongly regular matrices revisited

We prove a necessary condition and a sufficient condition for an n×n-matrix A with determinantal rank ρ(A)=t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its tth compound matrix Ct(A). In particular, a non-zero n×n-matrix A with ρ(A)=t over a local commutative ring R is strongly regular if and only if Tr(Ct(A)) is a unit in R, and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras.

GENERALIZED REED-MULLER CODES OVER GALOIS RINGS

A Galois ring may be considered as a common generalization of a finite field and a prime power integer residue ring. The generalized Reed-Muller codes over finite fields were introduced by Kasami et al. and the generalized Reed-Muller codes over prime power integer residue rings were constructed by M. Bhaintwal and S. K. Wasan. In this paper, we give an unifying approach for these constructions.

Influence of thermally grown oxides on interfacial friction during hot deformation of large-size forging ingots

High-strength steels are pre-heated in gas-fired furnaces before undergoing the open-die forging process. This process increases thermal oxidation on steel surfaces, affecting the interfacial friction between ingot and anvils during deformation. Two medium carbon high-strength steels with different nickel contents were oxidized, and the mechanical characteristics of oxide layers were investigated by micro and nano-indentation methods. It was found that the formed layers on high nickel steel had lower Young modulus and hardness compared to the steel with lower nickel. Finite element modeling and ring tests were used to assess oxide layers' effect on interfacial friction during deformation. The results demonstrated that oxide layers' formation decreased the interfacial friction and deformation load, acting as lubricants at high temperatures.

Counting points on diagonal equations over Galois rings GR(p2,2n)

Let R=GR(p2,2n) be a Galois ring and Fq a finite field, where q=p2n. In this paper, we study the number of solutions of equations of the form a1x1d1+⋯+asxsds=b with xi∈GR(p2,2ti), where ai,b∈R∖{0} and ti|n for all i=1,...,s, and employ results on quadratic forms over finite fields to give an explicit formula for the number of solutions of diagonal equations with restricted solution sets satisfying certain natural restrictions on the exponents. As a consequence, we present conditions for the existence of solutions.

Polyfunctions over commutative rings

A function [Formula: see text], where [Formula: see text] is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative [Formula: see text]. Based on this notion, we introduce ring invariants which associate to [Formula: see text] the numbers [Formula: see text] and [Formula: see text], where [Formula: see text] is the subring generated by [Formula: see text]. For the ring [Formula: see text] the invariant [Formula: see text] coincides with the number theoretic Smarandache or Kempner function [Formula: see text]. If every function in a ring [Formula: see text] is a polyfunction, then [Formula: see text] is a finite field according to the Rédei–Szele theorem, and it holds that [Formula: see text]. However, the condition [Formula: see text] does not imply that every function [Formula: see text] is a polyfunction. We classify all finite commutative rings [Formula: see text] with unit element which satisfy [Formula: see text]. For infinite rings [Formula: see text], we obtain a bound on the cardinality of the subring [Formula: see text] and for [Formula: see text] in terms of [Formula: see text]. In particular we show that [Formula: see text]. We also give two new proofs for the Rédei–Szele theorem which are based on our results.

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of FpFp[θ]-linear codes, where p∈{2;3} and θ≠θ2=0 that provides LCD codes over Fp.

Bounds for Coding Theory over Rings

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov’s weight over the integers modulo 2s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight.

Dynamics of magnetic flux propagation in bulk, single grain superconducting rings during pulsed field magnetisation

When used as trapped field magnets (TFMs), single grain, bulk high-temperature superconducting (HTS) rings are promising candidates for the generation of strong, uniform magnetic fields for nuclear magnetic resonance. The pulsed field magnetisation (PFM) technique provides a low cost, compact and portable method to magnetise these samples as TFMs; however it has proven difficult to achieve high trapped fields in HTS rings using PFM. To date, a record field of only 0.60 T has been achieved for rings magnetised by single-pulse PFM—compared with over 4 T for disc-shaped HTS—and the reasons for this discrepancy are poorly understood. In this work, we use the finite element method to model the propagation of magnetic flux into HTS rings under quasi-static zero field cooled magnetisation and PFM, and validate the results analytically and experimentally. Magnetic flux is found to penetrate finite HTS rings from both the inner and outer surfaces, inducing a negative field at the inner face of the ring. This field is reversed as the applied field increases past the point of full penetration, locally dissipating magnetic energy and heating the sample. HTS rings are therefore more susceptible to local instabilities that severely limit their ability to trap a useful magnetic field. Consequently, thermomagnetic stability of HTS rings during single-pulse PFM can only be ensured by taking careful consideration of reducing flux movement through the bulk around the point at which the field is reversed. This may require more advanced PFM techniques like waveform control or multi-pulse stepwise-cooling to reduce local heating and increase the trapped field.

LCP of group codes over finite Frobenius rings

LCP of group codes over finite Frobenius rings

$${{\,\mathrm{\texttt {VIC}}\,}}$$-modules over noncommutative rings

For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an application, we prove a very general twisted homology stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.

New Family of Polyphase Sequences with Low Correlation from Galois Rings

New Family of Polyphase Sequences with Low Correlation from Galois Rings

A Unique Representation of Cyclic Codes over GR(pn,r)

Let R be a Galois ring, GR(pn,r), of characteristic pn and of order pnr. In this article, we study cyclic codes of arbitrary length, N, over R. We use discrete Fourier transform (DFT) to determine a unique representation of cyclic codes of length, N, in terms of that of length, ps, where s=vp(N) and vp are the p-adic valuation. As a result, Hamming distance and dual codes are obtained. In addition, we compute the exact number of distinct cyclic codes over R when n=2.

Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree 2 n -1-n of the polynomial algebra 𝔽 2 [x 1 ,...,x n ], considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of GL n (𝔽 2 ). We generalize this result to all finite fields by studying a family of finite quotient rings R n,k , k∈ℕ * , of 𝔽 q [x 1 ,...,x n ], where each R n,k is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of R n,k , we also show that the space of indecomposable elements of 𝔽 q [x 1 ,...,x n ] in degree q n-1 -n has dimension equal to that of a complex cuspidal representation of GL n (𝔽 q ), that is (q-1)(q 2 -1)⋯(q n-1 -1).

Butson full propelinear codes

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the k{mathrm{th}} roots of unity, we can construct a larger Butson matrix over the ell mathrm{th} roots of unity for any ell dividing k, provided that any prime p dividing k also divides ell . We prove that a {mathbb {Z}}_{p^s}-additive code with p a prime number is isomorphic as a group to a BH-code over {mathbb {Z}}_{p^s} and the image of this BH-code under the Gray map is a BH-code over {mathbb {Z}}_p (binary Hadamard code for p=2). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.

Polynomial functions over dual numbers of several variables

Let [Formula: see text] be a positive integer. For a commutative ring [Formula: see text], the ring of dual numbers of [Formula: see text] variables over [Formula: see text] is the quotient ring [Formula: see text], where [Formula: see text] is the ideal generated by the set [Formula: see text]. This ring can be viewed as [Formula: see text] with [Formula: see text], where [Formula: see text] for [Formula: see text]. We investigate the polynomial functions of [Formula: see text] whenever [Formula: see text] is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on [Formula: see text] depending on the order of the pointwise stabilizer of the subring of constants [Formula: see text] in the group of polynomial permutations of [Formula: see text]. Further, we show that the stabilizer group of [Formula: see text] is independent of the number of variables [Formula: see text]. Moreover, we prove that a function [Formula: see text] on [Formula: see text] is a polynomial function if and only if a system of linear equations on [Formula: see text] that depends on [Formula: see text] has a solution.

The wiener index of the zero-divisor graph for a new class of residue class rings

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo p s q t for all distinct primes p, q and . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph Γ E (R) are derived for .

A Nonlinear Rotordynamics Model for Automotive Turbochargers Coupled to a Physical Model for a (SEMI) Floating Ring Bearing System

Abstract Automotive turbochargers (TCs) use an engine oil lubricated bearing system to produce acceptable performance (as per the engine volumetric efficiency) and proven reliability. However, the bearings also cause TC rotordynamic responses that are rich in subsynchronous whirl motions through reaching stable limit cycles. The paper describes the lubrication model for a finite length semi-floating ring bearing (SFRB) system and its coupling to the rotor and ring structure models for prediction of both linear and nonlinear system responses and their characterization in terms of motion amplitudes and whirl frequency content. The SFRB model includes a thermal energy transport network for the inner and outer films in both radial bearings and the thrust bearings located on the end sides of the ring. The large temperature difference between the hot shaft and a cold housing induces a three-dimensional thermal gradient in the fluid films and the floating ring, further exacerbated by the heat generated from drag power losses in the inner films adjacent to the rotor. The temperature gradients affect the lubricant viscosity and the bearing system operating clearances. The integration of the rotor and bearing system (RBS) equations of motion accounts for the SFRB nonlinear forces and starts from static equilibrium position, if existing. The model predictions of nonlinear behavior are accurate when benchmarked to a set of measurements procured in a gas stand test rig. The analysis also investigates the influence of the bearing inner clearance and rotor mass imbalance distribution on the onset, persistence and severity of SSV.

QSCs from the Euclidean sums of cyclic codes over finite rings

QSCs from the Euclidean sums of cyclic codes over finite rings

Extensions of Schönemann's theorem in Galois rings

Let p be a prime and Fp be a finite field of p elements. For a1,...,ak,b∈Fp, let NFp(a1,...,an,b) denote the number of the solutions (x1,...,xn)∈Fpn to the following linear equationsa1x1+⋅⋅⋅+anxn=b with the constraint xi≠xj for all i≠j. In 1839, Schönemann obtained the exact formula of NFp(a1,...,an,0) provided that ∑i=1nai=0 and ∑i∈Iai≠0 for all I⫋{1,...,n}. Schönemann's theorem was generalized by Bibak, Kapron and Srinivasan from Fp to Z/mZ for any positive integer m. Recently, Li and Yu generalized Schönemann's theorem to the finite field Fq with general prime power q. In this article, we give a further extension of Schönemann's theorem in the Galois rings GR(p2,r).