scholarly journals Skew Polynomial Extensions over Zip Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-9 ◽  
Author(s):  
Wagner Cortes
Keyword(s):  
Polynomial Extension ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
The Relationship ◽  
Polynomial Extensions

In this article, we study the relationship between left (right) zip property of and skew polynomial extension over , using the skew versions of Armendariz rings.

A GENERALIZATION OF NIL-ARMENDARIZ RINGS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350001 ◽  
Author(s):  
MOHAMMAD HABIBI ◽  
RAOUFEH MANAVIYAT
Keyword(s):  
Sufficient Conditions ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Monoid Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Special Cases ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
The Relationship

Let R be a ring, M a monoid and ω : M → End (R) a monoid homomorphism. The skew monoid ring R * M is a common generalization of polynomial rings, skew polynomial rings, (skew) Laurent polynomial rings and monoid rings. In the current work, we study the nil skew M-Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials to skew monoid rings. We resolve the structure of nil skew M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil skew M-Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. We consider central idempotents which are invariant under a monoid endomorphism of nil skew M-Armendariz rings and classify how the nil skew M-Armendariz rings behaves under various ring extensions. We also provide rich classes of skew monoid rings which satisfy in a condition nil (R * M) = nil (R) * M. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and those of the skew monoid ring R * M.

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 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.

scholarly journals Alpha - Skew Pi - Armendariz Rings

2018 ◽  
Vol 4 (1) ◽  
pp. 17
Author(s):  
Areej M Abduldaim ◽  
Ahmed M Ajaj
Keyword(s):  
Semiprime Ring ◽  
Important Goal ◽  
Identification Scheme ◽  
Modern Algebra ◽  
General Properties ◽  
Armendariz Rings ◽  
Armendariz Ring ◽  
The Relationship ◽  
Novel Algorithm

In this article we introduce a new concept called Alpha-skew Pi-Armendariz rings (Alpha - S Pi - AR)as a generalization of the notion of Alpha-skew Armendariz rings.Another important goal behind studying this class of rings is to employ it in order to design a modern algorithm of an identification scheme according to the evolution of using modern algebra in the applications of the field of cryptography.We investigate general properties of this concept and give examples for illustration. Furthermore, this paperstudy the relationship between this concept and some previous notions related to Alpha-skew Armendariz rings. It clearly presents that every weak Alpha-skew Armendariz ring is Alpha-skew Pi-Armendariz (Alpha-S Pi-AR). Also, thisarticle showsthat the concepts of Alpha-skew Armendariz rings and Alpha-skew Pi- Armendariz rings are equivalent in case R is 2-primal and semiprime ring.Moreover, this paper proves for a semicommutative Alpha-compatible ringR that if R[x;Alpha] is nil-Armendariz, thenR is an Alpha-S Pi-AR. In addition, if R is an Alpha - S Pi -AR, 2-primal and semiprime ring, then N(R[x;Alpha])=N(R)[x;Alpha]. Finally, we look forwardthat Alpha-skew Pi-Armendariz rings (Alpha-S Pi-AR)be more effect (due to their properties) in the field of cryptography than Pi-Armendariz rings, weak Armendariz rings and others.For these properties and characterizations of the introduced concept Alpha-S Pi-AR, we aspire to design a novel algorithm of an identification scheme.

scholarly journals Skew polynomial rings over σ-skew Armendariz rings

2016 ◽  
Vol 3 (1) ◽  
pp. 1183287
Author(s):  
V.K. Bhat ◽  
Meeru Abrol ◽  
Igor Yakov Subbotin
Keyword(s):  
Polynomial Rings ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Skew Polynomial

A new class of non-semiprime quasi-Armendariz rings

2014 ◽  
Vol 51 (2) ◽  
pp. 165-171
Author(s):  
Mohammad Habibi
Keyword(s):  
Polynomial Ring ◽  
Trivial Extension ◽  
Sufficient Condition ◽  
Nilpotent Ideal ◽  
Noncommutative Ring ◽  
Polynomial Extension ◽  
New Class ◽  
Monoid Ring ◽  
Armendariz Rings

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

scholarly journals RINGS WITH THE BEACHY–BLAIR CONDITION

2012 ◽  
Vol 11 (01) ◽  
pp. 1250006 ◽  
Author(s):  
ELENA RODRÍGUEZ-JORGE
Keyword(s):  
Power Series ◽  
Left Ideal ◽  
Zip Ring ◽  
Skew Polynomial ◽  
The Relationship

A ring satisfies the left Beachy–Blair condition if each of its faithful left ideal is cofaithful. Every left zip ring satisfies the left Beachy–Blair condition, but both properties are not equivalent. In this paper we will study the similarities and the differences between zip rings and rings with the Beachy–Blair condition. We will also study the relationship between the Beachy–Blair condition of a ring and its skew polynomial and skew power series extensions. We give an example of a right zip ring that is not left zip, proving that the zip property is not symmetric.

Sign in / Sign up

Export Citation Format

Share Document