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.     

Köthe’s upper nil radical for modules

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.     

Reduced multiplication modules

An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of Spec(M)\mbox{\rm Spec}(M). Also a finitely generated module M is a Baer module if and only if Spec(M)\mbox{\rm Spec}(M) is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N, Ann(x) \not í (N:M).\mbox{\rm Ann}(x) \not \subseteq (N:M). When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM.     

On the annihilators and attached primes of top local cohomology modules

Let ${\mathfrak{a}}$ be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}$ , where ${T_R(\mathfrak{a}, M)}$ is the largest submodule of M such that ${{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}$ . Several applications of this result are given. Among other things, it is shown that there exists an ideal ${\mathfrak{b}}$ of R such that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}$ . Using this, we show that if ${ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}$ , then ${{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}$ These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).     

有限域上幂剩余正规元的存在性

GF(q)是q个元的有限域,q是素数的方幂,n是正整数,GF(q~n)为GF(q)的n次扩张.用指数和估计的方法给出了3种情形下幂剩余正规元存在的充分条件,即(1)GF(q~n)中存在元ξ为GF(q)上的幂剩余正规元;(2)GF(q~n)中存在元ξ与ξ~(-1)同时为GF(q)上幂剩余正规元;(3)对GF(q~n)~*中任意给定的非零元a和b,GF(q~n)中存在元ξ与ξ~(-1)同时为GF(q)上d次幂剩余正规元,且满足Tr(ξ)=a,Tr(ξ~(-1))=b.     

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 $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

On 2-Systoles of Hyperbolic 3-Manifolds

We investigate the geometry of π ₁-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any ${\epsilon > 0}$ , if the manifold M has sufficiently large systole sys₁(M), the genus of any such surface in M is bounded below by ${{\rm exp}((\frac{1}{2} - \epsilon){\rm sys}_1(M))}$ . Using this result we show, in particular, that for congruence covers M _i → M of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of π ₁-injective surfaces satisfies ${{\rm log} \, {\rm sysg}(M_i) \gtrsim \frac{1}{3} {\rm log} \, {\rm vol}(M_i) ; (b)}$ there exist such sequences with the ratio Heegard ${{\rm genus}(M_i)/{\rm sysg}(M_i) \gtrsim {\rm vol}(M_i)^{1/2}}$ ; and (c) under some additional assumptions π ₁(M _i ) is k-free with ${{\rm log} \, k \gtrsim \frac{1}{3}{\rm sys}_1(M_i)}$ . The latter resolves a special case of a conjecture of Gromov.     

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

The Cycle Discrepancy of Three-Regular Graphs

Let G = (V, E) be an undirected graph and C(G){{\mathcal C}(G)} denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as

L7（3）与GL7（3）的OD-刻画

对于任意一个有限群G,令π（G）表示由它的阶的所有素因子构成的集合.构建一种与之相关的简单图,称之为素图,记作Γ（G）.该图的顶点集合是π（G）,图中两顶点p,g相连（记作p～q）的充要条件是群G恰有pq阶元.设π（G）={P₁,p2,…,p_x}.对于任意给定的p∈π（G）,令deg（p）:=|{q∈π（G）|在素图Γ（G）中,p～q}|,并称之为顶点p的度数.同时,定义D（G）:=（deg（p₁）,deg（p₂）,…,deg（p_s））,其中p₁2<…相似文献   

Matrix near-rings and 0-primitivity

In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}${\mathbb{M}_n(R) (n >1 )} and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and \mathbbM_n(R){\mathbb{M}_n(R)} has the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)}-subgroups, then the 0-primitivity of \mathbbM_n(R){\mathbb{M}_n(R)} implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)} is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.     

关于图的符号边全控制数

Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.     

形式三角矩阵环的PS性质和CESS性质

设$R$是环. 称右$R$-模$M$是PS-模,如果$M$具有投射的socle. 称$R$是PS-环,如果$R_R$是PS-模. 称$M$是CESS-模,如果$M$的任意具有基本socle的子模是$M$的某个直和因子的基本子模.本文给出了形式三角矩阵环 $T=\left( \begin{array}{cc} A & 0 \\     

Zeros of Monomial Brauer Characters

Let G be a finite group and p be a fixed prime. A p-Brauer character of G is said to be monomial if it is induced from a linear p-Brauer character of some subgroup(not necessarily proper) of G. Denote by IBr_m(G) the set of irreducible monomial p-Brauer′characters of G. Let H = G′O~p′(G) be the smallest normal subgroup such that G/H is an abelian p′-group. Suppose that g ∈ G is a p-regular element and the order of gH in the factor group G/H does not divide |IBr_m(G)|. Then there exists ? ∈ IBr_m(G) such that ?(g) = 0.     

A graph associated with the p\pi-character degrees of a group

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.     

A Remark on the Norm of Integer Order Favard Spaces

For a generator $A$ of a $C_0$-semigroup $T(\cdot)$ on a Banach space $X$ we consider the semi-norm $M^{k}_x:=\limsup_{t\to 0+}\|t^{-1}(T(t)-I)A^{k-1}x\|$ on the Favard space ${\cal F}_{k}$ of order $k$ associated with $A$. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities for bounded linear operators on ${\cal D}(A^k)$ can be extended to the larger space ${\cal F}_{k}$ by using the semi-norm $M^{k}_{(\cdot)}$. We also show that $M^{k}_{(\cdot)}$ is a norm equivalent to the norms that are usually considered in the literature if A has a bounded inverse.     

On a property of Abelian groups related to direct sums and products

Let T be an infinite set of prime numbers, $ \mathcal{M} $ be a set of groups $ \left\{ {\left. {\mathbb{Z}(p)} \right|p \in T} \right\} $ . An Abelian group A is said to be $ \mathcal{M} $ -large if $$ {\text{Hom}}\left( {A,\;\mathop { \bigoplus }\limits_{p \in T} \mathbb{Z}(p)} \right) = {\text{Hom}}\left( {A,\;\prod\limits_{p \in T} {\mathbb{Z}(p)} } \right). $$ This paper presents a characterization of $ \mathcal{M} $ -large torsion-free and mixed groups.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

Über die Bedingung Going up für {R \subset \hat{R}}

Let (R,\mathfrak m){(R,\mathfrak m)} be a noetherian, local ring with completion [^(R)]{\hat{R}} . We show that R ì [^(R)]{R \subset \hat{R}} satisfies the condition Going up if and only if there exists to every artinian R-module M with Ann_R(M) ì \mathfrakp{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with Ann_R(U)=\mathfrakp.{{\rm {Ann}}_R(U)=\mathfrak{p}.} This is further equivalent to R being formal catenary, to α(R) = 0 and to H^d_{\mathfrakq/\mathfrakp}(R/\mathfrakp)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals \mathfrakp ì \mathfrakq \subsetneq \mathfrakm{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where d = dim(R/\mathfrakp){d = {\rm {dim}}(R/\mathfrak{p})}.     

Première valeur propre du laplacien, volume conforme et chirurgies

We define a new differential invariant a compact manifold by , where V _c (M, [g]) is the conformal volume of M for the conformal class [g], and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili invariant defined by . The proof relies on the study of the behaviour of when one performs surgeries on M.