scholarly journals Cyclic codes and minimal strong Gröbner bases over a principal ideal ring

2003 ◽  
Vol 9 (2) ◽  
pp. 237-249 ◽  
Author(s):  
G.H. Norton ◽  
A. Salagean
Keyword(s):  
Principal Ideal ◽  
Cyclic Codes ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Principal Ideal Ring ◽  
Ideal Ring

scholarly journals Strong Gröbner bases for polynomials over a principal ideal ring

2001 ◽  
Vol 64 (3) ◽  
pp. 505-528 ◽  
Author(s):  
Graham H. Norton ◽  
Ana Sǎlǎgean
Keyword(s):  
Principal Ideal ◽  
Gröbner Bases ◽  
Galois Ring ◽  
Grobner Bases ◽  
Principal Ideal Domain ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Principal Ideal Ring ◽  
Ideal Ring

Gröbner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Gröbner bases, also known as D-bases. Several authors have shown that strong Gröbner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Gröbner bases and strong Gröbner bases when A is a principal ideal ring. We also give algorithms for computing Gröbner bases and strong Gröbner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Gröbner basis over a finite-chain ring, for example a Galois ring.

Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

1994 ◽  
Vol 40 (5) ◽  
pp. 1654-1661 ◽  
Author(s):  
Xuemin Chen ◽  
I.S. Reed ◽  
T. Helleseth ◽  
T.K. Truong
Keyword(s):  
Minimum Distance ◽  
Cyclic Codes ◽  
Gröbner Bases ◽  
Grobner Bases

On (α + uβ)-constacyclic codes of length 4ps over 𝔽pm + u𝔽pm∗

2019 ◽  
Vol 18 (02) ◽  
pp. 1950023 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta ◽  
Thang M. Vo
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Unit Formula

For any odd prime [Formula: see text] such that [Formula: see text], the structures of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] [Formula: see text] are established in term of their generator polynomials. When the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of two constacyclic codes of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that the ambient ring [Formula: see text] is a principal ideal ring. From that, the structure, number of codewords, duals of all such [Formula: see text]-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text].

Special Principal Ideal Rings and Absolute Subretracts

1991 ◽  
Vol 34 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Eric Jespers
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Identity Mapping ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Theoretic Version

AbstractA ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

Explicit Formulas Describing: The Binary Operations Calculus In \(E_{a,b}\)

2011 ◽  
Vol 2 (3) ◽  
pp. 16
Author(s):  
Chillali Abdelhakim ◽  
Mohamed Charkani
Keyword(s):  
Elliptic Curve ◽  
Principal Ideal ◽  
Group Homomorphism ◽  
Explicit Formulas ◽  
Principal Ideal Ring ◽  
Binary Operations ◽  
Ideal Ring

In this work we study the elliptic curve over theartinian principal ideal ring \(A=\mathbf{F}_q[\epsilon]\), \((\epsilon^3=0)\). More precisely, we establish a group homomorphism betweens \((\mathbf{F}_q^2,+)\) and theabelian group \(E_{a,b}\) of elliptic curve. For cryptographyapplications, we give meany various explicit formulas describingthe binary operations calculus in \(E_{a,b}\).

When is every module with essential socle a direct sum of automorphism-invariant modules?

2019 ◽  
Vol 19 (10) ◽  
pp. 2050185
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Saboura Dolati Pish Hesari ◽  
Mehdi Khoramdel
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Essential Extension ◽  
Principal Ideal Ring ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Ideal Ring ◽  
Von Neumann Regular

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

scholarly journals A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

2019 ◽  
Vol 14 (2) ◽  
pp. 515-530
Author(s):  
Maria Francis ◽  
Thibaut Verron
Keyword(s):  
Principal Ideal ◽  
Gröbner Bases ◽  
General Setting ◽  
Polynomial Systems ◽  
Regular Sequence ◽  
Grobner Bases ◽  
Integral Domains ◽  
Unique Factorization Domain ◽  
Univariate Polynomials ◽  
Principal Ideal Domains

AbstractSignature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

QUASI-MORPHIC RINGS

2007 ◽  
Vol 06 (05) ◽  
pp. 789-799 ◽  
Author(s):  
V. CAMILLO ◽  
W. K. NICHOLSON
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Ring ◽  
Bezout Ring ◽  
Ideal Ring ◽  
Left And Right ◽  
Regular Rings

A ring R is called left morphic if R/Ra ≅ l (a) for each a ∈ R, equivalently if there exists b ∈ R such that Ra = l (b) and l (a) = Rb. In this paper, we ask only that b and c exist such that Ra = l (b) and l (a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bézout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring.

Rings Whose Modules are ⊕ -Cofinitely Supplemented

2013 ◽  
Vol 0 (0) ◽  
Author(s):  
Ergül Türkmen
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Proper Generalization

Abstract It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

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