强Prüfer环的同调刻画

设R是环,R的小finitistic维数定义为fPD(R)=sup{pd_RM|M∈FPR}.本文证明了:若R是连通的强Prüfer环,则fPD(R)≤1.也证明了若R是强Prufer环,M∈FPR,且M是Q-挠模,则pd_RM≤1.     

On the global dimension of an algebra

Let algebra R = Λ/P, where Λ is a free algebra over a field w. gl. dim R: = {min n ¦? R-modules X, Y, Tor _n+1 ^R (X, Y)=0}. In order that w. gl. dim R≤2n (w. gl. dim R≤2n+1), it is necessary and sufficient that, for any two ideals of algebra Λ, a left ideal A and a right ideal B, containing ideal P, the following equation holds: $$AP^n \cap P^n B = AP^n B + P^{n + 1} (AP^n B \cap P^{n + 1} = AP^{n + 1} + P^{n + 1} B).$$      

高维空间的一个Heilbronn型问题

本文研究了以下Heilbronn型问题:设S是欧氏空间按R~k 中由有限个点A_1,A_2,…,A_n组成的集合,令d(S)=min{A_iA_j|1≤i相似文献   

弱总体维数和模的自同态

本文用模的自同态,给出弱总体维数≤ｎ的环的特征,其中ｎ≥０．设Ｒ为环,部分地回答了下列问题：何时任意有限表现Ｒ－模Ｍ有无穷分解：０→Ｍ→Ｆ₀→Ｆ₁→…→Ｆ_n→…,其中每个Ｆ_i均是有限生成投射的,ｉ＝１,２,…？     

Homological Dimension of Coalgebras and Crossed Coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C ^* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C ^*.     

近三角剖分图的均衡二重少圈覆盖

近三角剖分图是一连通平面图，其内面均为三角形而其外面可能不是．令Ｇ为一具有ｎ个节点的近三角剖分图，Ｃ为 Ｇ的一个小圈二重覆盖（ＳＣＤＣ）^［２］．令(?)则Ｃ₀。称为Ｇ的均衡小圈二重覆盖．本文将证明：若Ｇ为外平面图，则 δ（Ｃ₀）≤ ２；否则δ（Ｃ₀）≤４。     

The flat dimensions of injective modules

Let R be a ring. Define r. IFD(R) as r.IFD(R)=sup{fdE/E is an injective right R—module}. The purpose of this paper is to investigate this “global” dimension.     

自相似集的Hausdorff测度与连续性

对集合F Rn,以dim F和Hdim F(F)分别表示F的Hausdorff维数和dim F维Hausdorff测度.设T=T(f1,...,fm)为Rn中的自相似集,即由相似压缩组成的迭代函数系统{f1...,fm)的吸引子.假如fi(T)∩fj(T)= (i≠j),那么,对任意ε>0,存在δ>0,若D=D(g1,...,gm)为Rn中的自相似集并且sup{||fk(x)-gk(x)||:||x||≤1,1≤k≤m}<δ,则1HdimT(T)-Hdim D(D)|<ε.     

The global dimension of ω-smash coproducts

We mainly study the global dimension of ω-smash coproducts. We show that if H is a Hopf algebra with a bijective antipode S _H , and C _ω ? H denotes the ω-smash coproduct, then gl.dim(C _ω ? H) ≤ gl.dim(C) + gl.dim(H), where gl.dim(H) denotes the global dimension of H as a coalgebra.     

树的最大特征值的上界的一个注记

设T是一个树,V是T的顶点集．记dv是υ∈V的度,△是T的最大顶点度．设υ∈V且dw=1．记k=ew+1,这里ew是w的excentricity．设δj′= max{dυ:dist(υ,w)=j},j=1,2,…,k-2,我们证明和这里μ1(T)和λ1(T)分别是T的Laplacian矩阵和邻接矩阵的最大特征值．特别地,记δo′=2．     

On divided commutative rings

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P^-1 = { x ? T : xP ? P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.     

Modules Satisfying the S-Noetherian Property and S-ACCR

Let A = k[x₁,..., x_n] be a standard graded k-algebra over an infinite field k. We assume A is Cohen-Macaulay and has Krull dimension one. Let e denote the multiplicity of A and r - 1 the postulation number of A. Let I be a homogeneous ideal in A of grade one. Let j(I) denote the smallest degree of a regular form in I. Let l (I) denote the smallest power of (x₁,... , x_n) contained in I. If μ(I) denotes the minimum number of generators of I, then we show μ(I) ≤ e + j(I) - max{r,l(I)}, We then show how Dubreil's Second Theorem follows easily from this inequality     

Control subspaces of minimal dimension and root vectors

We investigate the following characteristic of a linear operator A in a Banach space X: disc {inf(dim R′:R′?R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AⁿR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system \(\dot x = Ax + Bu\) with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.     

全有向图的幂敛指数

设D为有向图，T（D）为D的全有向图（Total-digraph),k(D)和p(D)分别为D的幂敛指数（Index of convergence)与周期（Period)，本文证明了。1，对任意非平凡有向图D,p(T(D))=1,k(T(D))≤max{2p(D)-1,2K(D) 1}，特别地，当D为本原有向图时，k(T(D))≤k(D) 1，当D不含有向圈时，k(T(D))=2k(D)-1；当D为有向圈Cn时，k(T(D))=2n-1.2。对任意非平凡强连通图D,k(T(D))≥Diam(D) 1。我们还证明了以上界是不可改进的最好界。     

Banach 空间中极大单调算子扰动的值域

设X为实Banach空间, T:D(T)(?)X→2X*为极大单调算子, C: D(T)(?)X→X*为有界算子(未必连续),而C(T+J)-1为紧算子．本文在上述假设条件下,通过附加一定的边界条件应用Leray-Schauder度理论研究了下述包含关系:0∈(T+C)(D(T)∩ BQ(0)),0∈(T+C)(D(T)∩ BQ(0));以及S(?)R(T+C), intS(?)intR(T+C)(其中S(?) X*);B+D(?)R(T+C),int(B+D)(?)intR(T+C)(其中 B(?)X*,D(?)X*)的可解性,得出了一些新的结论．     

关于w-linked扩环

Let R ■ T be an extension of commutative rings.T is called w-linked over R if T as an R-module is a w-module.In the case of R ■ T ■ Q 0 （R）,T is called a w-linked overring of R.As a generalization of Wang-McCsland-Park-Chang Theorem,we show that if R is a reduced ring,then R is a w-Noetherian ring with w-dim（R） 1 if and only if each w-linked overring T of R is a w-Noetherian ring with w-dim（T ） 1.In particular,R is a w-Noetherian ring with w-dim（R） = 0 if and only if R is an Artinian ring.     

S-内射模及S-内射包络

设R是环.设S是一个左R-模簇,E是左R-模.若对任何N∈S,有Ext_R~1(N,E)=0,则E称为S-内射模.本文证明了若S是Baer模簇,则关于S-内射模的Baer准则成立;若S是完备模簇,则每个模有S-内射包络;若对任何单模N,Ext_R~1(N,E)=0,则E称为极大性内射模;若R是交换环,且对任何挠模N,Ext_R~1(N,E)=0,则E称为正则性内射模.作为应用,证明了每个模有极大性内射包络.也证明了交换环R是SM环当且仅当T/R的正则性内射包e(T/R)是∑-正则性内射模,其中T=T(R)表示R的完全分式环,当且仅当每一GV-无挠的正则性内射模是∑-正则性内射模.     

Endomorphism Algebras of 2-term Silting Complexes

We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D ^b (A) of A, the global dimension of \(\text {End}_{{D^b(A)}}(\mathbf {P})\) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D ^b (A) admits a 2-term silting complex P with \(\mathrm {gl. dim~}\text {End}_{{D^b(A)}}(\mathbf {P})\) infinite.