Extensions of Generalized Armendariz Rings

2006 ◽  
Vol 13 (02) ◽  
pp. 253-266 ◽  
Author(s):  
Chan Yong Hong ◽  
Tai Keun Kwak ◽  
S. Tariq Rizvi
Keyword(s):  
Polynomial Ring ◽  
Skew Polynomial Ring ◽  
Strong Connection ◽  
Principal Ideals ◽  
Ring Endomorphism ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

A ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. Such rings have been extensively studied in literature. For a ring endomorphism α, we introduce the notion of α-Armendariz rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established, and connections of properties of an α-Armendariz ring R with those of the ring R[x; α] are investigated. In particular, among other results, we show that there is a strong connection of the Baer property and the p.p.-property (principal ideals are projective) of the two rings, respectively. Several known results follow as consequences of our results.

On α-nilpotent elements and α-Armendariz rings

2015 ◽  
Vol 14 (05) ◽  
pp. 1550064
Author(s):  
Hong Kee Kim ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Hidetoshi Marubayashi
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250079 ◽  
Author(s):  
A. ALHEVAZ ◽  
A. MOUSSAVI
Keyword(s):  
Polynomial Rings ◽  
Trivial Extension ◽  
Skew Polynomial Ring ◽  
Matrix Rings ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Skew Polynomial ◽  
Armendariz Ring

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

ON THE ALGEBRAIC PARAMETERS OF CONVOLUTIONAL CODES WITH CYCLIC STRUCTURE

2006 ◽  
Vol 05 (01) ◽  
pp. 53-76 ◽  
Author(s):  
HEIDE GLUESING-LUERSSEN ◽  
BARBARA LANGFELD
Keyword(s):  
Polynomial Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Convolutional Codes ◽  
Complete Characterization ◽  
Field Size ◽  
Cyclic Structure ◽  
Skew Polynomial Ring ◽  
Principal Ideals ◽  
Skew Polynomial

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [4] that only certain combinations of the algebraic parameters (field size, length, dimension, and Forney indices) can occur for such cyclic codes. We will investigate whether all these combinations can indeed be realized by a suitable cyclic code and, if so, how to construct such a code. A complete characterization and construction will be given for minimal cyclic codes. It is derived from a detailed investigation of the units in the skew-polynomial ring.

Nilpotent elements and McCoy rings

2012 ◽  
Vol 49 (3) ◽  
pp. 326-337 ◽  
Author(s):  
Liang Zhao ◽  
Xiaosheng Zhu ◽  
Qinqin Gu
Keyword(s):  
Polynomial Ring ◽  
The Other ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Other Hand ◽  
Armendariz Rings ◽  
Skew Polynomial

We introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concepts of McCoy rings and nil-Armendariz rings. It is proved that every semicommutative ring is nil-McCoy. We shall give an example to show that nil-McCoy rings need not be semicommutative. Moreover, we show that nil-McCoy rings need not be right linearly McCoy. More examples of nil-McCoy rings are given by various extensions. On the other hand, the properties of α-McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x] are also investigated. For a monomorphism α of a ring R, it is shown that if R is weak α-rigid and α-reversible then R is α-McCoy.

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

A remark on skew factorization and new 𝔽4-linear codes

2021 ◽  
Author(s):  
Padmapani Seneviratne
Keyword(s):  
Polynomial Ring ◽  
Linear Codes ◽  
Skew Polynomial Ring ◽  
Complete Set ◽  
New Codes ◽  
Skew Factorization ◽  
Skew Polynomial ◽  
New Formula

Nine new [Formula: see text] linear codes with lengths [Formula: see text] and [Formula: see text] that improve previously best known parameters are constructed. By modifying these codes, another 17 new codes are obtained. It is conjectured that the complete set of factors of [Formula: see text] can be derived from the factors of [Formula: see text] for even values of [Formula: see text] in the skew polynomial ring [Formula: see text]. It is further shown that the [Formula: see text] code obtained is linearly complementary dual.

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

2014 ◽  
Vol 57 (3) ◽  
pp. 609-613 ◽  
Author(s):  
Alireza Nasr-Isfahani
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Base Ring ◽  
Necessary And Sufficient ◽  
Skew Polynomial

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

The theory of modules of separably closed fields 1

2002 ◽  
Vol 67 (3) ◽  
pp. 997-1015 ◽  
Author(s):  
Pilar Dellunde ◽  
Françoise Delon ◽  
Françoise Point
Keyword(s):  
Polynomial Ring ◽  
Quantifier Elimination ◽  
Additive Functions ◽  
Skew Polynomial Ring ◽  
Closed Fields ◽  
Skew Polynomial

AbstractWe consider separably closed fields of characteristic p > 0 and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the p-component functions.

scholarly journals A Characterization of separable polynomials over a skew polynomial ring

1985 ◽  
Vol 38 (2) ◽  
pp. 275-280
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Subject Classification ◽  
Skew Polynomial Ring ◽  
Skew Polynomial

AbstractThe characterization of a separable polynomial over an indecomposable commutative ring (with no idempotents but 0 and 1) in terms of the discriminant proved by G. J. Janusz is generalized to a skew polynomial ring R [ X, ρ] over a not necessarily commutative ring R where ρ is an automorphism of R with a finite order. 1980 Mathematics subject classification (Amer. Math. Soc.): 16 A 05.

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