Armendariz Rings And Extensions Of Armendariz Rings

Rege and Chhawchharia gave the definition of Armendariz ring in 1997. Following their definition, a ring R is called an Armendariz ring, if for any fg= 0 leads to aibi= 0 for all 0≤i≤ n and 0≤j≤m. We name the property mentioned in this definition "Armendariz condition". Such a ring is called Armendariz ring, since Armendariz proved, in 1974, that reduced ring satisfies Armendariz condition, i.e., reduced ring is an Armendariz ring. Rege and Chhawchharia gave many examples of Armendariz rings and non-Armendariz rings. They also defined several special rings related to Armendariz ring. They started the study of Armendariz rings. In 1998, Anderson and Camillo also did some further research work on Armendariz rings and gave many deep results.With the research on Armendariz ring going on, many mathematicians work on the generalizations of Armendariz ring and many definitions related to Armendariz ring ap-pear. In 2003, Lee and Wong defined weak Armendariz ring. In 2006, Liu gave another definition of weak Armendariz ring. In 2007, Huh defined π- Armendariz ring. Antoine defined nil Armendariz ring in 2008. It is easy to see that these definitions are all gener-alizations of Armendariz rings. In 1998, Anderson and Camillo proved that a ring R is Armendariz if and only if R[x] is Armendariz. They also proved that R[x]/{xn) (n> 2) is Armendariz if and only if R is a reduced ring. The conclusion also holds for the weak Armendariz ring defined by Lee and Wong. In this thesis, we give the definition of "rel-atived Armendariz ring" which is very close to the definition of Armendariz ring. Let B be a subring of a ring R, if for any fg= 0 leads to aibi∈B for all 0≤ i≤n and 0≤j≤ m, then we say R is an B-Armendariz ring. It is easy to see that Armendariz ring and central Armendariz ring are both B-Armendariz ring. We also introduce the concept of minimal subring p to study Armendariz ring. The use of p much more simplified our discussion.For B-Armendariz rings, we proved the following theorems.Theorem 2.1.2. Let B is a subring of R. Then R is a B-Armendariz ring if and only if R[x] is a B[x]- Armendariz ring.This theorem generalized the work of Anderson and Agayev.For the minimal subring p of R, we getTheorem 2.2.1. p is an ideal of R.Theorem 2.2.3. Let I be an ideal of R such that R/I and I are both Armendariz rings. If R or I is semiprime, then R is an Armendariz rings.Theorem 2.2.6. ρ(R[x])-ρ(R)[x].Theorem 2.2.7. If we take R as an subring of R[x]/(xn), where n is an positive integer, then R[x]/(xn) is an R- Armendariz ring if and only if R is a reduced ring.Let R be a ring and M be a R-R-bimodule. Let T(R, M) denote the ring consists of the matrix of the following form T(R, M) is called the trivial extension of R through M. We getTheorem 2.2.8. ρ(T(R, M))= T(ρ(R), N), where N is the R-R-bimodule generated by and Cf denotes set consists of the coefficients of f(x).Theorem 2.2.9. Let R and S be rings with identities, M be a R-S-bimodule, andAnderson and Camillo proved in 1998 that for semiprime left and right Noetherian ring R, R is an Armendariz ring if and only if Q(R) is a reduced ring. Further, Kim and Lee proved that if it! is a von Neumann regular ring and there is a classical right quotient ring Q(R) of R, then R is Armendariz if and only if R is reduced, if and only if Q(R) is an Armendariz ring, if and only if Q(R) is reduced. In this thesis, we discuss, for semiprime ring R, the relation of the Armendariz property of R and the Armendariz property of quotient rings of R. We get the following theorems.Theorem 3.2.1. Let R be a semiprime ring. Then R is Armendariz if and only if Qs is Armendariz.Theorem 3.2.2. Let R be a prime Goldier ring with identity. Qr and Qmr are the same as above. Then the following statements are equivalent. (1) R is Armendariz; (2) Qr is Armendariz; (3) Qmr is Armendariz.We also discussed prime Armendariz rings and annihilators. We getTheorem 3.3.1. For a prime ring R, the following statements are equivalent.(1) R is a (1,1)- Armendariz ring;(2) R is a (1,n)- Armendariz ring for every positive integer n;(3) R is a (n,1)- Armendariz ring for every positive integer n;(4) the set of left annihilators of elements in R is linear ordered;(5) the set of right annihilators of elements in R is linear ordered;(6){r(a)|a ∈R} U{I(a)| a ∈ R} is linear ordered.We also discuss commutative Armendariz rings. We getTheorem 4.1.1. If G(I)={xi0yn-i0,xi1yn-i1，…,xisyn-is} where i0<i1<…< is, then R/I is a Armendariz ring if and only if it= i0+t for 1≤t≤s.Where G(I)={ui,…,up} is the unique minimal monomial set of generator.We discuss zero dimensional local Armendariz rings.定理4.2.1. Let R be a commutative Artinian local ring. IfR is an Armendariz ring, then diml C(0:Cm)≥diml(0:Cm)/(0:C), for all ideals C of R.Quotient rings of a Armendariz ring need not be Armendariz rings, so that people are curious about the properties of quotient rings. Let R be an integeral domain, if any two elements of R has a greatest common divisor in R, then R is called a GCD. We study the quotient rings of integral domain and GCD with respect to a principle ideal and get the following theorem.Theorem 4.3.2. Let R be GCD and F be a primitive polynomial over R. Then R[x]/(f) is an Armendariz ring.A ring R is called semicomrmutative if for any a, b in R, ab= 0 always implies aRb= 0. Commutative ring lies between reduced ring and Abel ring. But they are not quite closely related. In Chapter 5, we give examples of rings which are both Armendariz and semicommutative. Let denote a subring of M3(R). We get the following theorems.Theorem 5.2.1. Let R be a reduced ring and α,β,γ be compatible endomorphisms of R, then S3(R) is semicommutative.Theorem 5.2.2. Let R be a reduced ring and α,β,γ be compatible endomorphisms of R, then S3 (R) is Armendariz.