On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

2019 ◽  
Vol 26 (02) ◽  
pp. 181-194 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Maximal Ideals

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

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].

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.

On a class of constacyclic codes of length 4ps over 𝔽pm + u𝔽pm

2019 ◽  
Vol 18 (02) ◽  
pp. 1950022 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta ◽  
Thang M. Vo
Keyword(s):  
Constacyclic Codes ◽  
Algebraic Structures ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Quadratic Polynomials ◽  
Direct Sum ◽  
Quotient Rings ◽  
Unit Formula

Let [Formula: see text] be a prime such that [Formula: see text]. For any unit [Formula: see text] of [Formula: see text], we determine the algebraic structures of [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], [Formula: see text]. If 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 an -[Formula: see text]-constacyclic code and an [Formula: see text]-constacyclic code of length [Formula: see text] If the unit [Formula: see text] is not a square, then [Formula: see text] can be decomposed into a product of two irreducible coprime quadratic polynomials which are [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. By showing that the quotient rings [Formula: see text] and [Formula: see text] are local, non-chain rings, we can compute the number of codewords in each of [Formula: see text]-constacyclic codes. Moreover, the duals of such codes are also given.

scholarly journals On some properties of⊕-supplemented modules

2003 ◽  
Vol 2003 (69) ◽  
pp. 4373-4387 ◽  
Author(s):  
A. Idelhadj ◽  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated ◽  
Principal Ideal Ring ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Ideal Ring

A moduleMis⊕-supplemented if every submodule ofMhas a supplement which is a direct summand ofM. In this paper, we show that a quotient of a⊕-supplemented module is not in general⊕-supplemented. We prove that over a commutative ringR, every finitely generated⊕-supplementedR-moduleMhaving dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ringRis semisimple if and only if the class of⊕-supplementedR-modules coincides with the class of injectiveR-modules. The structure of⊕-supplemented modules over a commutative principal ideal ring is completely determined.

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.

Commutative rings whose proper ideals are direct sum of completely cyclic modules

2016 ◽  
Vol 15 (09) ◽  
pp. 1650160 ◽  
Author(s):  
M. Behboodi ◽  
S. Heidari ◽  
S. Roointan-Isfahani
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Simple Formula ◽  
Commutative Rings ◽  
Index Set ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring

By two results of Köthe and Cohen–Kaplansky we obtain that “a commutative ring [Formula: see text] has the property that every [Formula: see text]-module is a direct sum of (completely) cyclic modules if and only if [Formula: see text] is an Artinian principal ideal ring” (an [Formula: see text]-module [Formula: see text] is called completely cyclic if each submodule of [Formula: see text] is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring [Formula: see text] is a direct sum of completely cyclic [Formula: see text]-modules if and only if [Formula: see text] is a principal ideal ring or [Formula: see text] is a local ring with maximal ideal [Formula: see text] such that there is an index set [Formula: see text] and a set of elements [Formula: see text] such that [Formula: see text] with each [Formula: see text] a simple [Formula: see text]-module and [Formula: see text] a principal ideal ring.

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 \(E_{a,b}\).

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.

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