On Monoid Rings Over Nil Armendariz Ring

2013 ◽  
Vol 42 (1) ◽  
pp. 1-21 ◽  
Author(s):  
A. Alhevaz ◽  
A. Moussavi
Keyword(s):  
Armendariz Ring

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 Radicals of Skew Inverse Laurent Series Rings

2016 ◽  
Vol 23 (02) ◽  
pp. 335-346
Author(s):  
A. Moussavi
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Laurent Series ◽  
Ring Extensions ◽  
Series Type ◽  
Armendariz Ring

Let R be a ring and α an automorphism of R. Amitsur proved that the Jacobson radical J(R[x]) of the polynomial ring R[x] is the polynomial ring over the nil ideal J(R[x]) ∩ R. Following Amitsur, it is shown that when R is an Armendariz ring of skew inverse Laurent series type and S is any one of the ring extensions R[x;α], R[x,x-1;α], R[[x-1;α]] and R((x-1;α)), then ℜ𝔞𝔡(S) = ℜ𝔞𝔡(R)S = Nil (S), ℜ𝔞𝔡(S) ∩ R = Nil (R), where ℜ𝔞𝔡 is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals.

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.

Further results on central Armendariz rings

2017 ◽  
Vol 16 (10) ◽  
pp. 1750194 ◽  
Author(s):  
Weixing Chen
Keyword(s):  
Prime Radical ◽  
Armendariz Rings ◽  
Armendariz Ring

A ring [Formula: see text] is central Armendariz if [Formula: see text] and [Formula: see text] [Formula: see text] over [Formula: see text] satisfy [Formula: see text] then all [Formula: see text] are central. It is proved that if [Formula: see text] is a central Armendariz ring, then [Formula: see text] implies that all [Formula: see text] are in its prime radical.

ON Π-NEAR-ARMENDARIZ RINGS

2009 ◽  
Vol 02 (01) ◽  
pp. 77-83
Author(s):  
Sh. Ghalandarzadeh ◽  
P. Malakooti Rad
Keyword(s):  
Basic Properties ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Armendariz Ring

In this note we introduce a concept, so-called π-Near-Armendariz ring, that is a generalization of both Armendariz rings and 2-primal rings. We first observe the basic properties of π-Near-Armendariz rings, constructing typical examples. We next extend the class of π-Near-Armendariz rings, through various ring extensions.

Reflexive rings and their extensions

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Liang Zhao ◽  
Xiaosheng Zhu ◽  
Qinqin Gu
Keyword(s):  
Quotient Ring ◽  
Weak Form ◽  
Polynomial Rings ◽  
Armendariz Ring

AbstractA right ideal I is reflexive if xRy ∈ I implies yRx ∈ I for x, y ∈ R. We shall call a ring R a reflexive ring if aRb = 0 implies bRa = 0 for a, b ∈ R. We study the properties of reflexive rings and related concepts. We first consider basic extensions of reflexive rings. For a reduced iedal I of a ring R, if R/I is reflexive, we show that R is reflexive. We next discuss the reflexivity of some kinds of polynomial rings. For a quasi-Armendariz ring R, it is proved that R is reflexive if and only if R[x] is reflexive if and only if R[x; x −1] is reflexive. For a right Ore ring R with Q its classical right quotient ring, we show that if R is a reflexive ring then Q is also reflexive. Moreover, we characterize weakly reflexive rings which is a weak form of reflexive rings and investigate its properties. Examples are given to show that weakly reflexive rings need not be semicommutative. It is shown that if R is a semicommutative ring, then R[x] is weakly reflexive.

Extensions of Generalized Reduced Rings

2005 ◽  
Vol 12 (02) ◽  
pp. 229-240 ◽  
Author(s):  
Chan Yong Hong ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak
Keyword(s):  
Polynomial Ring ◽  
Armendariz Ring

Anderson and Camillo studied the class of rings satisfying ZCn for n ≥ 2, which is a generalization of reduced rings. In this paper, we continue the study of such rings. We observe several extensions of rings satisfying ZCn. Rings satisfying the zero insertion property for n (simply, ZIn), which is a generalization of ZCn, are also introduced. In particular, we prove that every ring satisfying ZIn for some n ≥ 2 is a 2-primal ring. Furthermore, if R is an Armendariz ring satisfying ZIn for n ≥ 2, then the polynomial ring R[x] over R also satisfies ZIn.

scholarly journals Properties of Armendariz rings and weak Armendariz rings

2009 ◽  
Vol 85 (99) ◽  
pp. 131-137 ◽  
Author(s):  
Dusan Jokanovic
Keyword(s):  
Power Series ◽  
Direct Product ◽  
Nilpotent Ideal ◽  
Factor Ring ◽  
Ring Isomorphism ◽  
Armendariz Rings ◽  
Armendariz Ring

We consider some properties of Armendariz and rigid rings. We prove that the direct product of rigid (weak rigid), weak Armendariz rings is a rigid (weak rigid), weak Armendariz ring. On the assumption that the factor ring R/I is weak Armendariz, where I is nilpotent ideal, we prove that R is a weak Armendariz ring. We also prove that every ring isomorphism preserves weak skew Armendariz structure. Armendariz rings of Laurent power series are also considered.

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.

NIL-ARMENDARIZ RINGS AND UPPER NILRADICALS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250059 ◽  
Author(s):  
DA WOON JUNG ◽  
NAM KYUN KIM ◽  
YANG LEE ◽  
SUNG PIL YANG
Keyword(s):  
Regular Ring ◽  
Direct Method ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Armendariz Rings ◽  
Armendariz Ring

We continue the study of nil-Armendariz rings, initiated by Antoine, and Armendariz rings. We first examine a kind of ring coproduct constructed by Antoine for which the Armendariz, nil-Armendariz, and weak Armendariz properties are equivalent. Such a ring has an important role in the study of Armendariz ring property and near-related ring properties. We next prove an Antoine's result in relation with the ring coproduct by means of a simpler direct method. In the proof we can observe the concrete shapes of coefficients of zero-dividing polynomials. We next observe the structure of nil-Armendariz rings via the upper nilradicals. It is also shown that a ring R is Armendariz if and only if R is nil-Armendariz if and only if R is weak Armendariz, when R is a von Neumann regular ring.

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