【作者】 郭颖；

【导师】 杜现昆；

【作者基本信息】 吉林大学 ， 基础数学， 2004， 硕士

【摘要】 Armendariz环的研究是由Armendariz和Rege，Chhawchharia开始的，在文献[13]中，Rege和Chhawchharia于1997年引入了Armendariz环的概念，环R称为Armendariz环，如果对于任意的f（x）=sum fromn i=0 to m（a_ixⁱ，g（x））=sum from j=0 to n（b_jx^j）∈R[X]，当f（x）g（x）=0时，必有a_ib_j=0（0≤i≤m，0≤j≤n）．采用名称“Armendariz”是因为Armendariz早在1974年就已经发现reduced环（即不含非零幂零元的环）满足这个条件，即reduced环是Armendariz环。 显然，Armendariz环的子环以及Armendariz环的直积均为Armendariz环，从而Armendariz环的亚直积也是Armendariz环，但Armendariz环的商环却未必是Armendariz环，Rege和Chhawchharia证明了，PID的商环是Armendariz环。一般地，正如Anderson和Camillo所指出的，交换的主理想环是Armendariz环。更一般地，每个同态像都是Armendariz环的交换环恰为Gauss环。在本文中，我们证明了 定理2．4 设R是UFD，则对任意α∈R，商环R／（α）是Armendariz环。 定理2．4 推广了Rege和Chhawcharia 1997年一篇文章中的结果。 推论2．6 设R是UFD，f∈R[x₁，…，x_n]，则R[x₁，…，x_n]/（f）是Armendariz环。 一个环称为Abel环，如果它的每个幂等元都是中心的.ArlnendariZ;环是Abel环.R是Armefldariz环的另一个必要条件是:设R是Armendariz环，则当动=0，ac”b=O，a，乙，。任R，n全1时，必有ac6=0.Lee和、Vong证明了:设存在u，:任R满足沪二2少2=o，且。:二，)u笋。，则R不是Armendariz环. Anderson和Calnillo证明了R是Ar，ne，ldariz环补几{二}是Arll、en-dariz环;并证明了R!X{/（X“）（7、全2）是Ar;nendariz环铃几是red，，eed环.但是，Bl止phang和Rege证明了，对于四元数环H，HI刘/（护+l）不是Armendariz环.在本文中，我们证明了 推论2.12设R是Armendariz环且R:是redueed环，则R!二}/（xZ一l）是Armendariz环. Rege和Chhawchharia证明了任何环上的上三角形矩阵环从而全矩阵环都不是Armendariz环，因为它们都不是Abel环.如果一个上三角矩阵对角线上的元素完全相同，则称之为特殊的上三角矩阵，并称全体n阶特殊上三角矩阵构成的环为n阶特殊上三角矩阵环，记为S几（川.Kim和Lee证明了，若R是redueed环，则ST3（R）为Armendariz环，特另，1地，S几（R）是Armendariz环.Lee和WOng证明了 STZ（R）是Armel飞dariz环当且仅当R是redueed环.但是当n全4时，STn（R）不是Armerldariz环.对于Armendariz环R，S几（R）不必是Armendariz环.在本文中，我们证明了 定理2 .n设R是Armendariz环（不必有l）使得R:={二任川2x=0}是reduced环，则S（R）是Armendariz环，其中S（R）表示R上所有形的矩阵所构成的环.、、，.户/ L仃a a，I）Z声...，、、 口目 石父对于n全3，令呱（R）表示下列形式的矩阵所构成的环，、!/ BAaIo/了畜.‘、、其中a〔R，I是n一2阶单位矩阵，B是R上的（。一2）xZ矩阵，八是R上对角线元素为a的2阶上三角矩阵.阎占平证明了，若R是:、lllced环，则不玖（R）是A rmendariz环.在本文中，我们推广了这一结果，而且我们的证明也简单很多.我们证明了 定理3.3设a是环R的自同态，如果尺是a一rigid环，则呱（R）是a。一斜Armendariz环， Hirano证明了R是Armendariz环当且仅当环R的零化子理想和川刘的零化子理想之间有某种1一1对应关系.在本文中，我们将Hirallo的结果推广到了斜Armendariz环.我们证明了 定理&8设R是环，a是环R的自同构，则下面条件等价: （1）R是a一斜Armendariz环; （2）rR(明曰伙卜;。。(明(Ug卿是R的右零化子集到风筑叫的右零化子集的1一1对应; （3）扶（U）曰缅x;。](明(U红卿是R的左零化子集到川截a]的左零化子集的1一1对应. H ong，Kim和Kwak关于Baer环和p.p一环的结论可以作为我们的结果的推论直接得到.更多还原

【Abstract】 The study of Armendariz rings was initiated by Armendariz and Rege and Chhawchharia. In 1997, Rege and Chhawchharia introduced the notion of an Armendariz ring. A ring R is called Armendariz ring if whenever polynomials f(x) = atxl, g(x = bjX E R[X] satisfy f(x)g(x) = 0,we have aij = 0 ( 0 i m, 0 j n). The name " Armendariz ring " was chosen because Armendariz had noted that a reduced ring (i.e., rings without nonzero nilpotent elements) satisfies this condition, i.e., reduced rings are Armendariz rings.Obviously, subrings and direct products of Armendariz rings are Armendariz rings. Therefore, a subdirect product of Armendariz rings is also an Armendariz ring. But quotient rings of Armendariz rings need not be Armendariz rings. Rege and Chhawcharia proved that if R is a PID. then R/I is an Armendariz ring. Anderson and Camillo noted that a commutative PIR is an Armendariz ring. Generally, a ring whose every homornorphic image is an Armendariz ring is exactly a Gaussian ring. In this dessertation, we extend the result of Rege and Chhawchharia to a UFD.Theorem 2.4 Let R be a UFD, then R/(a) is an Armendariz ring for any a R.Corollary 2.6 Let R be a UFD, / e /2[a;i, , zn], then /?[.rt, . xn]/(f) is an Armendariz ring.A ring is called abelian if every idempotent of it is central. Armendariz rings are abelian. Another necessary condition for a ring jf to be an Armendariz ring is that ab = 0 and acnb - 0 imply that acb = 0 for a, /;, c 6 R and integer n 1. Lee and Wong showed that if there exist it, u R such that u2 = 0 = v’2 and uv = vu -- 0, then R, is not an Armendariz ring.Anderson and Camillo showed that a ring R is Armendariz if and only if R[x] is Armendariz, and that R[X]/(Xn) (n 2) is an Armendariz ring if and only if R is reduced. But Buhphang and Rege showed that for the division algebra of real quaternions H, //[o;]/(;r2 + 1) is not an Armendariz ring. In this dessertation, we proved thatCorollary 2.12 Let R be an Armendariz ring and _R2 = {x R\ 2x = 0} be reduced, then R[x}j( ?1) is an Armendariz ring.Rege and Chhawchharia showed that the ring of n x n upper triangular matrices over any ring, and hence an n x n matrix ring is not an Armendariz ring, because they are not abelian. An n x n matrix is called a specially upper triangular matrix if the elements of its diagonal are the same. We denote by STn(R) the ring consisting of n x n specially upper tiangular matrices. Kim and Lee showed STs(R) is an Armendariz ring if R is reduced. In particular, 5T2() is an Armendariz ring. Lee and Wong showed ST2(R) is an Armendariz ring if and only if R is reduced. But STn(R) is not an Armendariz ring for n 4. If R is an Armendariz ring, ST(R) is not necessarily an Armendariz ring. In this dessertation, we proved thatTheorem 2.11 Let R is an Armendariz ring(not necessarily with 1)vand let R2 = (x 6 R\ 2x - 0} be reduced. Then S(R) is an Arrnendarizring, where 5(7) denotes the ring consisting of the matrices b aFor n 3, let Wn(R) denotes the ring consisting of the matrices ofform’ al B0 Awhere o e R , /is the (n - 2) x (n - 2) identity matrix, B is an (n - 2) x 2 matrix over R, and A is a 2 x 2 upper triangular matrix over R with all diagonal entries equal to a. We proved thatTheorem 3.3 Let a be an endomorphism of a ring R. If P. is a-rigid, then Wn(R) is an oi-skew Armendariz ring.This theorem extends a result of Zhanping Yan, but also our proof is much simpler than his.Hirano showed that R is an Armendariz ring if and only if there is a 1-1 correspondence between the right (left respectively) annihilator ideals of R and those of R[x\. In this dessertation, we extend the result of Hirano to skew Armendariz rings. We proved thatTheorem 3.8 Let a be an automorphism of R. Then the following statements are equivalent:(1) R is an a-skew Armendariz ring;(2) TR(U) [x; a(U) (U C R) is a 1-1 correspondance from the set of right annihilators in R onto that in R[x; a};(3) IR(U) IR[X. a}(U) (U C R) is a. 1-1 correspondance from the setof left annihilators in更多还原