Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

环的Extensions扩张与余纯维数(英文)

Let R be a noetherian ring and S an excellent extension of R.cid(M) denotes the copure injective dimension of M and cfd(M) denotes the copure flat dimension of M.We prove that if M S is a right S-module then cid(M S)=cid(M R) and if S M is a left S-module then cfd(S M)=cfd(R M).Moreover,cid-D(S)=cid-D(R) and cfd-D(S)=cfdD(R).     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R~L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R~L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd_RX = GpdS(S?_R~L X). Similar result is obtained for Ding projective dimension of the S-complex S?_R~L X.     

Relative singularity categories with respect to gorenstein flat modules

Let R be a right coherent ring and D~b(R-Mod) the bounded derived category of left R-modules. Denote by D~b(R-Mod)_([G F,C]) the subcategory of D~b(R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K~b(F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D~b(R-Mod)_([G F,C])/K~b(F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules.     

广义Macaulay-Northcott模与Tor-群

Let （S,≤） be a strictly totally ordered monoid which is also artinian, and R a right noetherian ring. Assume that M is a finitely generated right R-module and N is a left Rmodule. Denote by [[MS,≤]] and [NS,≤] the module of generalized power series over M, and the generalized Macaulay-Northcott module over N, respectively. Then we show that there exists an isomorphism of Abelian groups：Tori[[ RS,≤]]（[[MS,≤]],[NS,≤]）≌ s∈S ToriR （M,N）.     

环的左(N,U)-凝聚维数和Excellent扩张

Let S be an excellent extension of a ring R and U a flat right S-module. We show in this paper that the left (N,U)-coherent dimension of S is equal to that of R.     

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].     

On Flat and Coflat Modules

It is well known that a ring R is left hereditary iff every left ideal of R is projective, iff every submodule of a projective module is projective (ld R≤1), iff every quotient module of an injective module is injective (1cd R≤1), where 1d R and 1cd R means the left global dimension resp. codimension of the ring R. These rings may be generalized to those of weak left global dimension at most 1 (wld R≤1). The latter condition holds iff every left ideal of R is flat, iff every submodule     

Left (N,U)-coherent Dimensions and Excellent Extensions of Rings

Let S be an excellent extension of a ring R and U a flat right 5-module. We show in this paper that the left (N, U)-coherent dimension of S is equal to that of R.     

The Auslander-type condition of triangular matrix rings

Let R be a left and right Noetherian ring and n,k be any non-negative integers.R is said to satisfy the Auslander-type condition G n (k) if the right flat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 i n 1.In this paper,we prove that R is Gn (k) if and only if so is a lower triangular matrix ring of any degree t over R.     

Tate Homology of Modules of Finite Gorenstein Flat Dimension

We introduce and investigate Tate homology $\widehat{{\mbox{\rm tor}}}$ of modules of finite Gorenstein flat dimension. In particular, we show that over a right coherent ring R, $\widehat{{\mbox{\rm tor}}}_{i}^{R}(M,N)\cong\widehat{{\mbox{\rm Tor}}}_{i}^{R}(M,N)$ for any right R-module M of finite Gorenstein projective dimension, any R-module N of finite Gorenstein flat dimension and any i?∈??. We also study the Tate homology $\widehat{{\mbox{\rm tor}}}$ of a cotorsion module of finite Gorenstein flat dimension in the paper.     

Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

由双边群相关Hopf模诱导的Morita等价

Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors （F3 = A  Bop A□ HBop -,G3 = （-）coH） between the categories A□HBopM and AMπB-H is a pair of inverse equivalences, when A is a faithfully flat π-H-Galois extension. Furthermore, the categories Moritaπ-H（A,B） and Morita □π-H（AcoH,BcoH） are equivalent, if A and B are faithfully flat π-H-Galois extensions.     

一类上三角矩阵环$W_{n}(p, q)$的斜Armendariz性质

设α是环R的一个自同态,称环R是α-斜Armendariz环,如果在R[x;α]中,(∑_(i=0)~ma_ix~i)(∑_(j=0)~nb_jx~j)=0,那么a_ia~i(b_j)=0,其中0≤i≤m,0≤j≤n.设R是α-rigid环,则R上的上三角矩阵环的子环W_n(p,q)是α~—-斜Armendariz环.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form

Ext_F-投射生成子与Ext_F-内射余生成子

设A是Abel范畴,F是Ext_A~1(-,-):A~(op)×A→A的加法子双函子.首先研究了Ext_(F-)投射生成子与Ext_F-内射余生成子的同调性质,其次引入了W_F-Gorenstein模的概念.特别地,证明了如果重复W_F-Gorenstein模的定义程序将不会产生新的模类.最后,统一并推广了许多参考文献中的结论.     

加权Dirichlet 空间上的一般Ces$\grave{a}$ro算子

对加权Dirichlet空间${\cal D}_{\alpha}=\left\{f\in H(D) ; ||f||_{{\cal D}_{\alpha}}^{2}=|f(0)|^{2}+\int_{D}|f'(z)|^{2}(1-|z|)^{\alpha}\d m(z)<+\infty \right\},~~-1<\alpha<+\infty,$我们研究了其上一般Ces$\grave{a}$ro算子的有界性. 此处$H(D)$表示复平面单位圆盘$D$上全纯函数的全体.     

Characterization of $2$-Primal Near-Rings

In 1999,Kim and Kwak asked one question that "Is a ring R 2-primal if OP■P for each P ∈ mSpec(R)?".In this paper,we prove that if O P has the IFP for each P ∈ mSpec(N),then OP■P for each P ∈ mSpec(N) if and only if N is a 2-primal near-ring and also we give characterization of 2-primal near-rings by using its minimal 0-prime ideals.     

${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解

本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.