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On weakened -skew Armendariz rings
No full textsummary:In this note, for a ring endomorphism and an -derivation of a ring , the notion of weakened -skew Armendariz rings is introduced as a generalization of -rigid rings and weak Armendariz rings. It is proved that is a weakened -skew Armendariz ring if and only if is weakened -skew Armendariz if and only if is weakened -skew Armendariz ring for any positive integer
Central Armendariz rings
No full textWe introduce the notion of central Armendariz rings which are a generalization of Armendariz rings and investigate their properties. We show that the class of central Armendariz rings lies strictly between classes of Ar- mendariz rings and abelian rings. For a ring R, we prove that R is central Armendariz if and only if the polynomial ring R[x] is central Armendariz if and only if the Laurent polynomial ring R[xx1-] is central Armendariz. Moreover, it is proven that if R is reduced, then R[x]/(xn) is central Armendariz, the converse holds if R is semiprime, where (xn) is the ideal generated by x n and n ? 2. Among others we also show that R is a reduced ring if and only if the matrix ring T n-2 n (R) is central Armendariz, for a natural number n ? 3 and k = [n/2]
Weak Quasi-Armendariz Rings
No full textIn this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/ is weak quasi-Armendariz, where R[x] is the polynomial ring over R and is the ideal of R[x] generated by x(n). Various properties of weak quasi-Armendariz rings are also observed.National Research Foundation of Korea (NRF); Ministry of Education, Science and Technology [2010-0022160]; Pusan National UniversityThe authors thank the referee for his/her very careful reading of the manuscript and very many suggestions that improved the paper. The third named author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0022160) and the fourth named author was supported by a 2-Year Research Grant of Pusan National University
?-Armendariz Rings and Related Concepts
No full textIn this paper we investigated some new properties of ?-Armendariz rings and studied the relationships between ?-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, ?-compatible rings and others. We proved that if R is a central Armendariz, then R is ?-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz?-compatible ring, then R is ?-Armendariz. Examples are given to illustrate the relations between concepts
RIGIDNESS AND EXTENDED ARMENDARIZ PROPERTY
No full textFor a ring endomorphism alpha of a ring R, Krempa called alpha a rigid endomorphism if a alpha(a) = 0 implies a = 0 for a is an element of R, and Hong et al. called R an alpha-rigid ring if there exists a rigid endomorphism alpha. Due to Rege and Chhawchharia, 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. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., alpha-Armendariz rings and alpha-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between alpha-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an alpha-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.Ministry of Education, Science and Technology [2010-0022160]The third named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(No. 2010-0022160)
On Armendariz rings
No full textÇalışmamızda Armendariz halkalar ve Armendariz halkalarla bağlantılı halkalar aralarındaki ilişkiler incelendi. Bu tez dört bölümden oluşmaktadır. İlk bölümde diğer bölümlerde kullanacağımız tanım ve teoremler verildi. İkinci bölümde Armendariz halkalarla bağlantılı polinom halkaları üzerinde Baer ve p.p.-halkalar incelendi ve Armendariz özelliğine sahip olmayan örnekler verildi. Bir R halkasının Armendariz olması için gerek ve yeter koşulun R[x] polinom halkasının Armendariz olması durumu gösterildi. Matris halkalarında Armendariz özeliğine sahip olan ve olmayan halkalar incelendi. Üçüncü bölümde Armendariz halkaları üzerinde polinom halkaları incelendi, yarı değişmeli halkaların özellikleri ile yarı değişmelilik ve Armendarizlik arasındaki ilişkiler gösterildi. Bundan başka Armendariz halkalarda genişlemeler araştırıldı. Son bölümde abelian halkalar çalışılarak abelian halkalar ile Armendariz halkalar arasındaki bağlantılar incelenmiştir.In this work, we studied relationship between Armendariz rings and the related rings. The study consists of four chapters. In the first chapter, we present some definations and theorems that will be used in the following chapters. In the second chapter, Baer and p.p.-rings were examined on polynomial rings which are related with Armendariz rings and examples were given for non Armendariz rings. 'A ring R is Armendariz if and only if polynomial ring R[x] is Armendariz' were shown . In addition Armendariz property on matrix rings, were examined . In the third chapter, polynomial rings on Armendariz rings were studied, moreover the relationships between semicommutativy and Armendariz property with semicommutative rings, were shown . Furthermore, extentions on Armendariz rings were investigated . In the last chapter, we studied abelian rings and its relations with the Armendariz rings are examined
Central Armendariz Rings
No full textWe introduce the notion of central Armendariz rings which are a
generalization of Armendariz rings and investigate their properties. We show
that the class of central Armendariz rings lies strictly between classes of Armendariz
rings and abelian rings. For a ring R, we prove that R is central
Armendariz if and only if the polynomial ring R[x] is central Armendariz if and
only if the Laurent polynomial ring R[x; xn] is central Armendariz. Moreover,
it is proven that if R is reduced, then R[x]=(xn) is central Armendariz, the
converse holds if R is semiprime, where (xn) is the ideal generated by xn and
n 2. Among others we also show that R is a reduced ring if and only if the
matrix ring Tn-2
n (R) is central Armendariz, for a natural number n 3 and
k = [n=2]
ARMENDARIZ AND QUASI-ARMENDARIZ SEMIRINGS AND PS SEMIRINGS
No full text In this paper we extend some results of ([2], [11], [12], [13], [15]) for non commutative semirings with identity 1 ≠ 0. We prove the following theorems: (1) Let R be a CN-semiring such that 0 is a P-primary ideal of R and P2 = 0. Then R is a quasi-Armendariz semiring. (2) Let R be a semiring, M an [R, R]-bisemimodule and R′ = R ⊕ M, the trivial extension of R by M. If R is a prime semiring, then R′ is a quasi-Armendariz (resp. p.s. quasi-Armendariz) semiring if and only if M is a quasi-Armendariz (resp. p.s. quasi-Armendariz) [R, R]-bisemimodule in the sense that for f ∈ R[x] ( resp . R⟦x⟧), g ∈ M[x] ( resp . M⟦x⟧) such that fRg = 0 implies that aiRbj = 0; and gRf = 0 implies that bjRai = 0 for each coefficient ai of f and bj of g. </jats:p
π-Armendariz Rings and Related Concepts
No full textIn this paper we investigated some new properties of π-Armendariz rings and studied the relationships between π-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, α-compatible rings and others. We proved that if R is a central Armendariz, then R is π-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz π-compatible ring, then R is π-Armendariz. Examples are given to illustrate the relations between concepts
Alpha - Skew Pi - Armendariz Rings
No full textIn 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
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