Armendariz Rings And Armendariz Property Of Some Quotient Rings

Armendariz rings were introduced by Rege and Chhawchharia in 1997.Since then Armendariz rings have attracted attentions of many researchers.There are several generalizations and many important results on Armendariz rings.A ring without nonzero nilpotent elements is called a reduced ring.Ar-mendariz discovered that a reduced ring R has the following property:for any polynomials f(x)=(?)aixi and g(x)=(?)bjxj,if f(x)g(x)=0,then Oibj=0(0?i?m,0?j?n).The property above is called the Armendariz property.Rege and Chhawchharia called rings with the Armendariz property Armendariz rings.Armendariz rings are a generalization of reduced rings.Let R be an Armendariz ring.It is natural to determine which relat-ed rings of R are Armendariz rings,for example,subrings,quotient rings,polynomial rings,matrix rings,etc.We first give a detailed survey of the re-search on the topic.Then we give the necessary and sufficient condition that R[X]/(I,xn)is an Armendariz ring,and finally a result on the quotient rings modulo annihilator ideals of Armendariz rings is generalized.The thesis is organized as follows.In the second section,we review properties of Armendariz rings.Section 3 is devoted to some Armendariz subrings of upper triangulax matrix rings.Section 4 surveys various generalizations of Armendariz rings.In section 5 we generalize results due to Anderson and Camillo about R[x]/(xn)and due to Lee and Wong about quotient rings modulo annihilator ideals.The main results are as follows:Theorem 5.3.Let I be an ideal of R and n? 2,Then R[x}/(I,xn)is an Armendariz ring if and only if R/I is a reduced ring.Theorem 5.5.If R is a ring and ki? 0,then R[x1,…,xn]/(x1k1,…,xnkn)is an Armendariz ring if and only if one of the following holds.1.R is an Armendariz ring and ki ?1 for all i.2.R is a reduce ring and ki>1 for only one i.Let S,T be non-empty subsets of ring R,M(S,T)=?r?srt=0,Vs?s,t?T} and I(S,T)=M(SR,RT).Theorem 5.8.If R is an Armendariz ring,then R/I(S,T)is an Armen-dariz ring.Theorem 5.9.If R is an Armendariz ring,and if S,T are non-empty subsets of ring then R/I?(S,T)is an Armendariz ring for any ordinal num-ber ?.