【作者】 杨柳；

【导师】 杜现昆；

【作者基本信息】 吉林大学 ， 基础数学， 2017， 博士

【摘要】 无非零幂零元的环称为约化环(或简约环).Armendariz最先发现约化环R满足下述条件:对R上的任意多项式f(x)= 0 a1 + aax十…十gmxm,b(x)= +0十十…n,当f(x)g(x)= 0时,必有aibj= 0,0 ≤ i≤m,0 ≤ j ≤ n.受此启发,Rege和Chhawchharia研究了满足上述条件的环,并称之为Armendariz环.从此以后,Armendariz环及其各种推广得到了广泛研究.我们将把上述条件称为Armendariz条件.Anderson和Camillo利用Armendariz环给出了 Gauss环的刻画,Hirano利用Armendariz环描述了 R与R[x]中左零化子之间的关系.从Armendariz环的定义可以看出,Armendariz环的子环是Armendariz环.但是,Armendariz环的商环未必是Armendariz环.因此,有两个自然的问题:1.Armendariz环的哪些扩环仍是Armendariz环?2.Armendariz环的哪些商环仍是Armendariz环?本文讨论群环、平凡扩张、矩阵子环、多项式代数关于单项式理想的商环的Armendariz条件.在第二章,我们讨论了域上群代数与一般群环的Armendariz性质.证明了在大多数情况下,群代数是Armendariz环当且仅当它是约化环.这表明,刻画Armendariz群代数本质上等价于解决群代数的零因子问题.所谓零因子猜测是指:域上的扭自由群代数是约化的,这是群环理论中的著名难题.对于一般群环,我们的研究表明循环群的群环和四元数环是研究群环Armendariz性质的关键.我们还讨论了 Hamilton四元数除环上群环的Armendariz性质.设K8表示8阶四元数群.定理2.2.12.设R是2-扭自由的、约化的、交换环.则下述命题等价.1.RK8 是 Armendariz 环;2.RK8是约化环;3.x2 + y2 + z2 = 0在R中没有非零解.定理2.2.21.设F是域,G是扭群或扭自由群.则FG是Armendariz环当且仅当或者FG是约化环,或者chF = p>0,G是扭Abbel群,并且G的p-分量是循环群或拟循环群.定理2.2.22.1.群G的复群代数CG是Armendariz环当且仅当G的所有特征为0群代数都是Armendariz环.此时,G的扭子群T是Abel群.2.群G的所有群代数都是Armendariz环当且仅当G在所有有限域上的群代数都是Armen-dariz 环.设T是群G的有限阶元素之集,并设T是G的子群.令△(G,T)=RG(1-T).定理2.3.5.设R是2-扭自由的.则RK8是Armendariz环当且仅当四元数环H(R)是Armendariz环.定理2.3.8.若RG是Armenndariz 环,则1.T是G的子群;2.△(G,T)是 Armendariz 环;3.R(G/T)是 Armendariz 环.反之,若上述条件成立,并且｜T｜是无限的或者｜T｜有限且在R中可逆,则RG是Armendariz环,设H是四元数除环.定理2.4.5.设T是扭群,则群环HT是Armendariz环当且仅当T是初等Able 2-群.定理2.4.7.设n是正整数,q ∈H,则H[x]/(xn+q)是Armendariz环当且仅当下列条件之一成立(1)q = 0;(2)q 不是实数,(3)n = 1;(4)n = 2 且 q 是负实数.定理2.4.9.设f(x)是实系数多项式,且degf = ≥ 1,则H[x]/(f(x))是Armendariz环当且仅当f(x)在实数域中有n个根(计重数).在第三章,我们讨论广义矩阵子环(特别是平凡扩张的子环)的Armendariz性质.给出了平凡扩张的子环的构造,讨论了平凡扩张的子环是Armendariz环的充分条件和必要条件,推广了已有的结果.定理3.1.5.设M是R-双模.给定M的子双模K及导子d:R → M/K,令则Td,K是平凡扩张R ∝ M的子环且满足π(Td.K)=R.反之,R ∝ M的满足π(T)=R的子环T都可这样构造.设Φ:A → B是环的映射(不必是同态).如果对于任意满足b1b2 = 0的b1,b2 ∈ B都存在a1,a2 ∈AA使得Φ(a1)=)，Φ(a2)= b2,a1a2=0,则称Φ保零积.定理3.2.2.设有保零积的环同态Φ:A[x]→ B[x]使得Φ(A)(?)B且Φ(x)= x.若A是Armendariz环,则 B 是 Armendariz环.定理3.2.3.设AB是两个环且B是Armendariz环.如果映射Φ:A → B保零积,则其扩张映射Φ:A[x]→ B[x]也保零积.定理3.2.7.设T是R ∝ M的子环且π(T)=R.1.如果 T 是 Armendariz 环且 Annr(Annl(M0))= M0,则 R是 Armendariz 环;2.如果T是Armendariz环,π｜T保零积,则对于满足fg=0的f,g∈R[∈ 恒有｜fMg ∈∩Mf(-g)｜ = 1.3.如果R是Armendariz环,M是Armendariz双模,并且对于满足f分=0的f,分∈ R[x]恒有fMg ∩ Mf(-g)=0,则 T 是 Armendariz 环.定理3.2.8.设T是R∝M的子环使得π(T)= R且 π｜T保零积.假设R是Armendariz 环,M是 Armendariz 双模.如果 T 是 Armendariz 环,则 R ∝ M0 是 Armendariz 环.设M是R双模,σ,T是R的自同态.记aσ = σ(a),a ∈R.令R ∝TσM = R×M且有乘法如下贝R ∝Tσ M是有单位元的环.我们刻画了 R ∝Tσ M是Armendariz环的充分必要条件.定理3.3.2.设R是环,M是R-双模,σ,T是R的自同态.则R ∝Tσ M是4rmendariz环的充分必要条件为:1.R 是 Armendariz 环;2.M 是 Armendariz(Rσ,Rτ)-双模;定理3.3.5.设σ:→ A和τ:R → B是环的满同态,M是(A,B)-双模且通过纯量限制视为R-双模,令则T(R,σ,T,M)是 Armendariz 环当且仅当R ∝ M 是 Armendariz 环,其中R=R/(kerσ∩ker T).设α是R的自同态,如果对于任意a,b ∈ R均有aα(b)= 0当且仅当ab = 0,则称α是R的相容自同态.设n ≥ 2是任意正整数,令其中=[n/2],即当n为偶数时,n = 2k;当n为奇数时,= 2k+ 1.定理3.4.6.若R是约化环,α1,α2,…,αn是R的相容自同态.则在Un(R)的主对角元上依次作用α1α玖2,.,αn,所得到的矩阵环的子环Sn(R)是Armendariz环.在第四章,我们讨论了K...,xd]/I的Armendariz性质,其中K是域,I是单项式理想.设I是R的理想,如果理想商环R/I是Armendariz环,则称I是Armendariz理想,简称A-理想.定理4.2.10.设I,J都是R的不可约单项式理想.假设IJ,J(?)I,且不全是A-理想.那么In J是A-理想当且仅当I,J是如下情形之一:在下面的几个定理中,花括号下方数字表示单项式的次数.定理4.3.1.设k为非负整数.若G(I)是下列三种情形之一,mjI是A-理想..定理4.3.2.若I的极小生成集为则I是A-理想.定理4.3.4.设c>2,k,l≥0.若I的极小生成集是下列情形之一,则I是A-理想:定理4.3.5.设c>1,q>3.若I有如下形式的极小生成集,则I是A-理想:定理4.4.6.设I是A-理想,则T(I)内行(列)距不超过6的两个格点的整点凸包必含于Γ(I).更多还原

【Abstract】 A ring is called reduced if it has no nonzero nilpotent elements.Armendariz noted that a reduced ring R has the following property:for any polynomials f(x)= a0+a1x+…+amxm and g(x)= b0+b1x + …+bnxn over a ring R,f(x)g(x)= 0 0 implies aibi=0,0≤i≤m,0≤j≤n.Motivated by this fact,Rege and Chhawchharia named a ring with the above property Armen-dariz and initiated the research on Armendariz rings.We call the above-mentioned condition as Armendariz condition.Anderson and Camillo characterized Gaussian rings in terms of Armendariz rings.Hirano characterized the relation between the left annihilators of R and the left annihilators of R[x]by Armendariz rings.From the definition of Armendariz ring,we see that subrings of an Armendariz ring are Armendariz,but quotient rings of an Armendariz ring may not be Armendariz.So two natural questions arise:1.Which extension rings of Armendariz rings are Armendariz?2.Which quotient rings of Armendariz rings are Armendariz?In this thesis,we discuss the Armendariz property of group rings,trivial extensions,subrings of matrix rings,the factor rings of polynomial algebras modulo monomial ideals.In the second chapter,we discuss the Armendariz property of group algebras over a field and general group rings.We prove that in most cases,a group algebra is Armendariz if and only if it is reduced,which means that to characterize an Armendariz group algebra is equivalent to discussing the zero divisor conjecture of group algebras.The zero divisor conjecture says:a group algebra of a torsion-free group over a field is reduced.It is a famous difficult problem in group ring theory.For general group rings,our research indicates that group rings of cyclic groups and quaternion rings are crucial in the research of group rings.We also discuss the Armendariz property of the group ring over Hamilton’s quaternion division ring.Let K8 denote the quaternion group of order 8.Theorem 2.2.12.Let R be a 3-torsion-free reduced commutative ring.Then the following conditions are equivalent:1.RK8 is Armendariz;2.RK8 is reduced;3.x2+ y2+ z2 = 0 has no nonzero solution in R.Theorerm 2.2.21.Let F be a field and G be a torsion or torsion-free group.Then FG is Armendariz if and only if either FG is reduced,or chF = p>0 and G is a torsion abelian group whose p-component is cyclic or quasi-cyclic.Theorerm 2.2.22.Let G be a group.1.The complex group algebra CG over G is Armendariz if and only if each group algebra over G of characteristic 0 is Armendariz.In this case,the torsion subgroup T of G is Abelian.2.Each group algebra over G is Armendariz if and only if each group algebra over G and any finite field is Armendariz.Let T be the subset of G consisting of all elements of finite order in G.Assume T is a subgroup of G.Write △(G,T)= RG(1-T).Theorem 2.3.5.Suppose R is 2-torsion-free.Then RK8 is Armendariz if and only if the quaternion ring H(R)is Armendariz.Theorerm 2.3.8.If RG is an Armendariz ring,then1.T is a subgroup of G;2.△(G,T)is an Armendariz ring;3.R(G/T)is an Armendariz ring.Conversely,if the conditions above hold and ｜T｜ is infinite or ｜T｜ is finite and i-nvertible in R,then RG is an Armendariz ring.Let H denote Hamilton’s quaternion division ring.Theorem 2.4.5.Let T be a torsion group.The group ring HT is Armendariz if and only if T is an elementary Abelian 2-group.Theorem 2.4.7.Let n be a positivc integer and g E H.Then H[x]/(Xn + q)is Armendariz if and only if one of the following conditions holds:(1)q = 0;(2)q(?)R;(3)n = 1;(4)n = 2,g G R and g<0.Theorem 2.4.9.Let f(x)be a polynomial with real coefficients and deg f = n ≥ 1.Then H[x]/(f(x))is Armenda-riz if and only if f(x)has n roots in R(counting multiplicities).In the third chapter,we discuss the Armendariz property of subrings of generalized matrix(especially,subrings of the trivial extension).We give the construction of subrings of the trivial extension,and give sufficient conditions and necessary conditions for subrings of the trivial extension to be Armendariz,which generalize the existing results.Theorerm 3.1.5.Let M be an R-bimodule.Given a sub-bimodzule K of M and derivative d:R→M/K,letThen Td,K is a subring of the trivial extension R ∝ M and π(Td,k)=R.Conversely,every subring T of R ∝ M with π(T)= R can be constructed in this way.Let Φ:A → B be a map between rings(not necessarily a ring homomorphism).If for each pair b1,b2 ∈ B with b1b2 = 0,there exist a1,a2 ∈ A such that Φ(α1)= b1,Φ(a2)= b2,a2 = 0,then called Φ reserving zero-product.Theorem 3.2.2.Let Φ:A[x]→B[x]be a reserving zero-product ring homomorphism such thatΦ(A)(?)B and Φ(x)= x.If A is Armendariz,then B is Armendariz.Theorem 3.2.3.Let A,B be rings and let B be an Armendariz ring.If a map Φ:A → B reserves zero-product,then the extension map Φ:A[x]→B[x]also reserves zero-product.Theorem 3.2.7.Let T be a subing of R∝ M,and π(T)= R.1.If T is Armendariz,and Annr(Ann,(M0))= MO,then R is Armendariz;2.If T is Armendariz and π｜T reserves zero-product,then ｜fMg∩f(-g)｜ = 1 for all f,g ∈ R[x]with f g = 0.3.If R is an Armendariz ring,M is an Armendariz bimodule,and fMg ∩Mf(-g)=0 for all f,g ∈R[x]with fg = 0,then T is an Armendariz ring.Theorem 3.2.8.Let T be a subring of R ∝ M such that π(T)= R and π｜T reserves zero-product.Suppose R is an Armendariz ring and M is an Armendariz bimodule.If T is Armendariz,then R ∝ M0 is Arme∽dariz.Let M be an R-bimodule,σ,T endomorphisms of R and aσ =σ(a)for a ∈ R.Set R ∝ M =R x M with multiplication defined as followsThen R ∝Tσ M is a ring with identity.We present a sufficient and necessary condition for R ∝Tσ M to be Armendariz.Theorem 3.3.2.Let R be a ring,M an R-bimodule,and σ,T endomorphisms of R.Then a sufficient and necessary condition for R ∝Tσ M to be an Armendariz ring is.R is an Armendariz ring;2.M is an Armendariz(Rσ,RT)-bimodule;Theorem 3.3.5.Let σ:R → A and T:R → B be epimorphisms of rings,and let M be an(A,B)-bimodule that is viewed as an R-bimodule via the restriction of scalars.Let Then T(R,σ,T,M)is an Armendariz ring if and only if R ∝ M is an Armendariz ring,in which R = R/(ker σ∩ ker T).Let α be an endomorphism of R.Then α is called compatible if aα(b)= 0 is equivalent to ab = 0 for all a,b ∈ R.For a positive integer n>2,letwhere k =[n/2],which means n = 2k when n is even and n = 2k + 1 when n is odd.Theorem 3.4.6.If R is a reduced ri’rng witlh compatible erndomorphisms α1,α2,...,and Sn(R)denotes the subring of the matrix ring obtained by imposing α1,α2,...,αn on diagonal elements of Un(R)in sequence.Then Sn(R)is Armendariz,In the forth chapter,we discuss the Armendariz property of K[x1,...,Xd]/I,in which K is a field and I is a monomial ideal.Let I be an ideal of R.If the factor ring R/I is Armendariz,then we say that I is an Armendariz ideal,briefly an A-ideal.Theorem 4.2.10.Let I and J be irreducible monomial ideals of R.Suppose I(?)J,J(?)I,and I and J are not all A-ideals.Then I∩ J is an A-ideal if and only if one of the following conditions holds.In the following theorems,the numbers under the braces denote the orders of the monomials.Theorem 4.3.1.Let k be a nonnegative integer.Then an ideal I is an A-ideal if its minimal generating set G(I)is one of the following three situationsTheorem 4.3.2.If the minimal generating set of I isthen I is an A-ideal.Theorem 4.3.4.Let c>2 and k,l ≥ 0,Then I is an A-ideal if the minimal generating set of I is one of the following situationsTheorem 4.3.5.Let c>1 and q>3.Then I is an A-ideal if the mi-nimal generating set of I isTheorem 4.4.6.Let I be A-ideal,then G(xiyi,xi1yi1)(?)C(I)for all xiyi,xi1,yi1 ∈I with min{｜i—i’｜,｜j-j’｜}≤6.更多还原