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.

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.

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

2018 ◽  
Vol 25 (04) ◽  
pp. 595-618 ◽  
Author(s):  
Kamal Paykan ◽  
Abasalt Bodaghi
Keyword(s):  
Power Series ◽  
Triangular Matrix ◽  
Inverse Power ◽  
Direct Sums ◽  
Matrix Rings ◽  
Series Ring ◽  
Semiprime Rings ◽  
Armendariz Rings ◽  
Power Series Rings ◽  
Series Type

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, [Formula: see text]-quasi-Armendariz), is introduced and studied. It is shown that the [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime [Formula: see text]-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

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.

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.

On app skew generalized power series rings

2013 ◽  
Vol 50 (4) ◽  
pp. 436-453
Author(s):  
A. Majidinya ◽  
A. Moussavi
Keyword(s):  
Power Series ◽  
Left Ideal ◽  
Commutative Monoid ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Armendariz Ring ◽  
Left Annihilator

By [12], a ring R is left APP if R has the property that “the left annihilator of a principal ideal is pure as a left ideal”. Equivalently, R is a left APP-ring if R modulo the left annihilator of any principal left ideal is flat. Let R be a ring, (S, ≦) a strictly totally ordered commutative monoid and ω: S → End(R) a monoid homomorphism. Following [16], we show that, when R is a (S, ω)-weakly rigid and (S, ω)-Armendariz ring, then the skew generalized power series ring R[[S≦, ω]] is right APP if and only if rR(A) is S-indexed left s-unital for every S-indexed generated right ideal A of R. We also show that when R is a (S, ω)-strongly Armendariz ring and ω(S) ⫅ Aut(R), then the ring R[[S≦, ω]] is left APP if and only if ℓR(∑a∈A ∑s∈SRωs(a)) is S-indexed right s-unital, for any S-indexed subset A of R. In particular, when R is Armendariz relative to S, then R[[S≦]] is right APP if and only if rR(A) is S-indexed left s-unital, for any S-indexed generated right ideal A of R.

RADICALS OF SKEW GENERALIZED POWER SERIES RINGS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250129 ◽  
Author(s):  
A. R. NASR-ISFAHANI
Keyword(s):  
Power Series ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Ordered Monoid ◽  
Monoid Homomorphism ◽  
Property Transfer ◽  
Power Series Rings ◽  
Armendariz Ring

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Ni ℓ*(R)[[S, ω]], where α is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip and 2-primal property, transfer between R and the skew generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz.

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