Skew Polynomial Rings Research Articles

Semilinear clannish algebras

AbstractWe define a class of associative algebras generalizing ‘clannish algebras’, as introduced by the second author, but also incorporating semilinear structure, like a skew polynomial ring. Clannish algebras generalize the well‐known ‘string algebras’ introduced by Butler and Ringel. Our main result is the classification of finite‐dimensional indecomposable modules for these new algebras.

Threshold Changeable Secret Sharing (TCSS) via Skew Polynomials

Secret sharing is a tool to divide a secret into multiple shares, so that to reconstruct the secret, it is necessary to collect several shares under certain conditions. Secret sharing was first introduced by Adi Shamir, the scheme is based on polynomials. The secret is represented as a constant value polynomial, and the points on the polynomial graph serve as shares. Secret reconstruction is performed through Lagrange interpolation at a minimum of k out of n points, known as a threshold scheme (k, n). In 2010, Zhang Y. designed a secret sharing scheme through skew polynomials. Involving the role of the automorphism σ in the skew polynomial ring increases the complexity in share distribution and secret reconstruction. In 2012, Zhang Z. designed a secret sharing scheme with a threshold that can change according to the participants present during the reconstruction process, known as Threshold Changeable Secret Sharing (TCSS). This aims to prevent external parties from pretending to be valid participants in order to learn the secret. In this research, a TCSS scheme will be designed using skew polynomials. The aim is to make the TCSS scheme's calculations more complex, making it harder for adversaries to access the secret.

Isomorphism between skew polynomial rings

Isomorphism between skew polynomial rings

A note on the Jacobson radical of Ore extensions

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R [ x ; α , δ ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation AX−XB=C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector spaces, and matrices over skew polynomial rings. The obtained results are applied to Sylvester-polynomial-conjugate matrix equations over complex numbers and quaternions. The main role in our approach is played by skew polynomial rings, which are well-known tools in algebra to provide examples of asymmetry between left-sided and right-sided versions of many ring objects.

Skew π-Baer polynomial rings

A ring [Formula: see text] with an automorphism [Formula: see text] and a [Formula: see text]-derivation [Formula: see text] is called [Formula: see text]-[Formula: see text]-Baer (respectively [Formula: see text]-Baer) if the right annihilator of every projection invariant [Formula: see text]-left ideal (respectively left ideal) of [Formula: see text] is generated by an idempotent, as a right ideal. We study the relationship between the [Formula: see text]-[Formula: see text]-Baer (respectively [Formula: see text]-Baer) property of a ring [Formula: see text] and those of the skew polynomial ring [Formula: see text], skew inverse power series rings [Formula: see text] and the skew inverse Laurent series ring [Formula: see text]. Examples are provided to illustrate and delimit the theory.

Fast Kötter–Nielsen–Høholdt interpolation over skew polynomial rings and its application in coding theory

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most O~sωM(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{O}}\\left( s^\\omega {\\mathfrak {M}}(n)\\right) $$\\end{document} operations in Fqm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_{q^m}$$\\end{document}, where n is the length of the code, s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s$$\\end{document} the interleaving order, M(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {M}}(n)$$\\end{document} the complexity for multiplying two skew polynomials of degree at most n, ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} the matrix multiplication exponent and O~·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{O}}\\left( \\cdot \\right) $$\\end{document} the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing.

Iterated Ore polynomial maps

We study a natural notion of polynomial maps attached to elements of an iterated Ore extension [Formula: see text]. We develop some tools to analyze these maps such as good points, multilinear transformations and P-independence. We also present polynomial maps arising in the Multivariate extensions [Formula: see text], introduced by Martínez-Peñas and Kschischang [Evaluation and interpolation over multivariate skew polynomial rings, J. Algebra 525 (2019) 111–139], through the use of Multilinear polynomial maps.

The Eliahou-Kervaire resolution over a skew polynomial ring

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.

Ozone Groups and Centers of Skew Polynomial Rings

Abstract We introduce the ozone group of a noncommutative algebra $A$, defined as the group of automorphisms of $A$, which fix every element of its center. In order to initiate the study of ozone groups, we study polynomial identity (PI) skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial ring to be Gorenstein (resp. regular) in low dimension.

On the algebraic structure of (M,σ,δ)-skew codes

In this paper, we are interested in linear codes invariant under the (σ,δ)-action of a given matrix M, that we call (M,σ,δ)-skew codes. Various subclasses of these codes, depending on the (σ,δ)-cyclical property of M, are derived. In particular, we distinguish between three fundamental types: (M,σ,δ)-cyclic, (M,σ,δ)-quasi-cyclic and (M,σ,δ)-multi-cyclic. These different subclasses allow us to obtain some of the well known types of linear codes as special cases depending on the choice of M. We observe that the (σ,δ)-similarity in Mn(Fq) gives rise to various relations between these codes. As common in the study of cyclic codes and their generalizations, we use the skew polynomial ring Fq[x,σ,δ] to study (M,σ,δ)-skew codes. As an application of the theory developed here, we construct some optimal linear codes and MDS codes over different alphabets with additional desirable properties such as self orthogonality and linear complementary duality (LCD).

A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation

Let R_r=Z_(2^m )+uZ_(2^m ) be a finite ring, where u^2=r for r∈Z_(2^m ), m is a positive integer, and m≥2. In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R_r with an automorphism θ_r and a derivation δ_(θ_r ). We generalize the skew-cyclic codes over Z_4+uZ_4; u^2=1 to the skew-cyclic codes over R_r, and call such codes as δ_(θ_r )-cyclic codes. We investigate the structures of a skew polynomial ring R_r [x,θ_r,δ_(θ_r ) ]. A δ_(θ_r )-cyclic code is showed to be a left R_r [x,θ_r,δ_(θ_r ) ]-submodule of (R_r [x,θ_r,δ_(θ_r ) ])/〈x^n-1〉 . We give the generator matrix of a δ_(θ_r )-cyclic code of length n over R_r. Also, we present the generator matrix of the dual of a free δ_(θ_r )-cyclic code of even length n over R_r.

Skew-polynomial-sparse matrix multiplication

Based on the observation that Q(p−1)×(p−1) is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for (p−1)×(p−1) matrix multiplication over Q, where p is a prime number. The algorithm has complexity O(Tω−2p2), where T≤p−1 is a parameter determined by the skew-polynomial-sparsity of input matrices and ω is the asymptotic exponent of matrix multiplication. Here a matrix is skew-polynomial-sparse if its corresponding skew polynomial is sparse. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity O∼(tω−2p2+p2log⁡1ν), where t≤p−1 is the skew-polynomial-sparsity of the product and ν is the probability parameter. The main feature of the algorithms is the acceleration for matrix multiplication if the input matrices or their products are skew-polynomial-sparse.

On weak annihilators and nilpotent associated primes of skew PBW extensions

We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier (Weak annihilator over extension rings, 2012) in the setting of skew PBW extensions. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research.

Critical and injective modules over skew polynomial rings

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on R, a criterion for S to have injective hulls of all simple S-modules locally Artinian - that is, for S to satisfy property (⋄). It is easy and well known that if α is of finite order, then S has this property, but in order to get the criterion when α has infinite order we found it necessary to classify all cyclic (Krull) critical S-modules in this case, a result which may be of independent interest. With the help of the above we show that Sˆ=k[[X]][θ,α] satisfies (⋄) for all k-algebra automorphisms α of k[[X]].

On the structure of ideals in a family of skew polynomial rings

On the structure of ideals in a family of skew polynomial rings

Division algebras and MRD codes from skew polynomials

AbstractLet $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring.Using skew polynomials $f\in R$ , we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

Extensions of i-reversible rings

A ring [Formula: see text] is said to be i-reversible if for every [Formula: see text] [Formula: see text][Formula: see text], [Formula: see text] is a nonzero idempotent implies [Formula: see text] is an idempotent. It is known that the rings [Formula: see text] and [Formula: see text] (the ring of all upper triangular matrices over [Formula: see text]) are not i-reversible for [Formula: see text]. In this paper, we provide a nontrivial i-reversible subring of [Formula: see text] when [Formula: see text] and [Formula: see text] has only trivial idempotents. We further provide a maximal i-reversible subring of [Formula: see text] for each [Formula: see text], if [Formula: see text] is a field. We then give conditions for i-reversibility of trivial, Dorroh and Nagata extensions. Finally, we give some independent sufficient conditions for i-reversibility of polynomial rings, and more generally, of skew polynomial rings.

(β,γ)-Skew QC Codes with Derivation over a Semi-Local Ring

In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=β(b)y+γ(b), where β is an automorphism and γ is a β-derivation on S so that S[y;β,γ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;β,γ] and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.