Classification of Hardy Submodules and Characteristic Space Theory（A Brief and Selective Survey）

This paper is a brief and selective survey on classification of Hardy Submodules over the polydisk. The survey reports some progress in classification of Hardy submodules.     

$QTAG$-Modules whose $h$-Pure-$S$-High Submodules Have Closure

A right module $M$ over an associative ring $R$ with unity is a $QTAG$-module if every finitely generated submodule of any homomorphic image of $M$ is a direct sum of uniserial modules. This article considers the closure of $h$-pure-$S$-high submodules of $QTAG$-modules. Here, we determine all submodules $S$ of a $QTAG$-module $M$ such that each closure of $h$-pure-$S$-high submodule of $M$ is $h$-pure-$\overline{S}$-high in $\overline{M}$. A few results of this theme give a comparison of some elementary properties of $h$-pure-$S$-high and $S$-high submodules.     

弱半素子模的一些刻划

引入n′-系的概念,且给出弱半素子模的两个等价条件:(i)设K是左R-模M的子模,则K是M的弱半素子模当且仅当C(K)=M\K是n′-系;(ii)设K是左R-模M的子模,则K是M的弱半素子模当且仅当对任意f∈HomR(R,M),及任意A R,若f(A2)K,就有f(A)K.     

On Modules Which are Isomorphic to Relatively Divisible or Pure Submodules of Each Other

Abstract

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub- modules of each other.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

Prime Submodules and Flat Modules

In this paper, some characterizations of prime submodules in flat modules and, particularly, in free modules are given. Furthermore, the height of prime submodules and some saturated chain of prime submodules are also given.     

遗传C~*-子代数与可补闭子模的关系定理

设 E是 Hilbert C~*-模,证明了E的每个闭子模均可补的充要条件是 K(E)的每个非零遗传C~*-子代数B,均有形式B=pK(E)p,其中p为L(E)中投影元。     

Prime Submodules and a Sheaf on the Prime Spectra of Modules

We define and investigate a sheaf of modules on the prime spectra of modules, and it is shown that there is an isomorphism between the sections of this sheaf and the ideal transform module.     

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.     

Closed Submodules of Holomorphic Functions with Two Generators

Let I be a closed submodule over a polynomial ring in a space of holomorphic functions on a domain in the complex plane. We establish sufficient conditions under which I is generated by two functions or two special submodules. As a corollary, it follows from these results that if an invariant subspace W C (a,b) (with respect to the differentiation operator) admits spectral synthesis, then it is the solution space of a system of two homogeneous convolution equations.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.     

Affine Configurations and Pure Braids

We show that the fundamental group of ordered affine-equivalent configurations with at least five points in the real plane is isomorphic to the pure braid group in as many strands, modulo its centre. In the case of four points, this fundamental group is free with 11 generators. This work was carried out at the Instituto de Matemáticas, UNAM and partially funded by the Sistema Nacional de Investigadores and a doctoral fellowship from CONACyT México.     

Krull Dimension and Classical Krull Dimension of Modules

Using the concept of prime submodule defined by Raggi et al. in [16 Raggi , F. , Rios , J. , Rincón , H. , Fernández-Alonso , R. , Signoret , C. ( 2005 ). Prime and irreducible preradicals . J. Algebra Appl. 4 ( 4 ): 451 – 466 .[Crossref], [Web of Science ®] , [Google Scholar]], for M ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that if M is progenerator in σ[M], τ ∈M-tors such that M has τ-Krull dimension then cl.K _τdim (M) ≤ k _τ(M). Also we show that if M is noetherian, τ-fully bounded, progenerator of σ[M], and M ∈ 𝔽_τ, then cl·K _τdim (M) = k _τ(M).     

Pure inductive limit state and Kolmogorov's property

Let be a C^*-dynamical system where be a semigroup of injective endomorphism and ψ be an (λ_t) invariant state on the C^* subalgebra and is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state canonically associated with ψ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state ψ to prove that Kolmogorov's property [A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003) 189–209] of the Markov semigroup is a sufficient condition for the inductive state to be pure. As an application of this criteria we find a sufficient condition for a translation invariant factor state on a one-dimensional quantum spin chain to be pure. This criteria in a sense complements criteria obtained in [O. Bratteli, P.E.T. Jorgensen, A. Kishimoto, R.F. Werner, Pure states on , J. Operator Theory 43 (1) (2000) 97–143; A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003) 189–209] as we could go beyond lattice symmetric states.     

On the Pure Injectivity of Certain Infinitely Generated Modules over Dedekind Pullbacks

Let R be a pullback of Dedekind domains onto a common quotient field. I discuss the pure injectivity of certain R-modules arising in a natural way as generalizations of indecomposable pure injective finitely generated R-modules.     

Prime and primary submodules of certain modules

In this paper we characterize all prime and primary submodules of the free R-module R ⁿ for a principal ideal domain R and find the minimal primary decomposition of any submodule of R ⁿ . In the case n = 2, we also determine the height of prime submodules.     

FBN Modules

For M ∈ R-Mod and τ ∈M-tors, we define the concept of fully τ-bounded module as a generalization of the concept of fully τ-bounded ring. We prove that for a τ-noetherian module M with local τ _M -Gabriel correspondence, which is a progenerator of σ[M] and with τ is FIS-invariant, then M is fully τ-bounded. Also, we show that if M is τ-noetherian and fully τ-bounded, then M has local τ _M -Gabriel correspondence.     

Cohomology of Pure Braid Groups of exceptional cases

Consider the ring R:=Q[τ,τ−1] of Laurent polynomials in the variable τ. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ. In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We compute this cohomology for the cases I₂(m), H₃, H₄, F₄ and An with 1?n?7.     

A Conjecture Concerning the Pure Exponential Diophantine Equation a^x ＋ b^y = c^z

Let a, b, c, r be fixed positive integers such that a^2 ＋ b^2 = c^r, min（a, b, c, r） 〉 1 and 2 r. In this paper we prove that if a ≡ 2 （mod 4）, b ≡ 3 （mod 4）, c 〉 3.10^37 and r 〉 7200, then the equation a^x ＋ b^y = c^z only has the solution （x, y, z） = （2, 2, r）.