Representations of Finite Posets Over Discrete Valuation Rings

Given a discrete valuation ring R, a non-negative integer m, and a finite partially ordered set S, the representation type of a category rep(S*, R, m) of finite rank free R-module representations of S is characterized in terms of S and m. As an application, representation types of some categories of torsion-free abelian groups of finite rank are determined.     

Induktive Limiten endlich erzeugter freier Moduln

If X is a set, R a ring and M an R-module, then it is shown that the R-module of all M-valued functions on X taking on X only finitely many values is free over R. This result is applied to show that inductive limits of finitely generated free R-modules are free over R, provided that the mappings of the inductive system are of a certain diagonal type. No restrictions are placed on the index set. There is a dual statement for projective limits. Among other applications it is shown that the -1^st Borel-Moore homology group (with values in any sheaf over an arbitrary integral domain and taken with respect to any paracompactifying family of supports) of a locally compact space vanishes.     

广义逆多项式模的co-Hopf性质

设R是结合环(可以没有单位元),(S,≤)是严格全序幺半群,序≤是Artin的且对任意s∈S,有0≤s,则对任意具有性质(F)的左R-模M,[MS,≤]是co-Hopf左[[RS,≤]]一模当且仅当M是co-Hopf左R-模.     

投射可迁环上矩阵环的若当同态

设R′是一个环,Mn′（R′）是R′上的n′×n′矩阵环.如果环R有不变基数性质并且每个有限生成的投射左R-模是自由模,则R是一个投射自由环.如果环R≌Mr（S）,其中S是一个投射自由环,则R是一个投射可迁环.当R是一个投射可迁环时,给出了从Mn′（R′）到Mn（R）（n′≥n≥2）的若当同态的代数公式.     

Noether环上的幂稳定自由模

设I是Noether环R的投射理想, I^m=Iⁿ, m≠n. 该文证明, 有限生成投射右R - 模幂稳定自由当且仅当(1) 存在环S使得I^|m-n|( S ( R且有限生成投射S - 模是幂稳定自由; (2) 有限生成投射右R/I^|m-n| - 模幂稳定自由.     

内射模的t－平坦性

设Ｒ为环，ｔ是左Ｒ－模范畴的一个遗传挠理论．文中证明了下述各点等价：（１）每个内射左Ｒ－模是ｔ－平坦的；（２）每个ｔ－有限表现左Ｒ－模的内射包络是ｔ－平坦的；（３）每个ｔ－有限表现左Ｒ－模是自由Ｒ－模的子模；（４）每个ｔ－有限表现左Ｒ－模是自反的且其对偶模是Ｈ－有限生成的．     

Free modules over hjelmslev rings in which not every maximal linearly independent subset is a basis

First it will be shown that every left-noetherian AH-ring is left-artinian. For an AH-ring R every finite linearly independent subset of a free left R-module V can be completed to a basis of V. But a maximal linearly independent subset of a free left R-module V need not be a basis of V. For an H-ring R, every maximal linearly independent subset of a free left R-module V is a basis of V if and only if the H-ring R is left-noetherian or V is finitely generated.     

Torsions in modules

In the category of right modules over the ring E=End_R(F), where F is a free right R-module, a torsion is defined. It is known as Tol'skaya torsion. The correlation between torsion-free E-modules in the sense of Tol'skaya and torsion-free E-modules in the sense of Bass is investigated. It is shown that the ring R is a right cogenerator if and only if in the ring of endomorphisms of any free R-module, r((J))=J for all finitely generated right ideals J.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 527–534, October, 1973.The author thanks L. A. Skornyakov for his valuable discussions and observations.     

FGF group rings

Periodica Mathematica Hungarica - A ring R is called a left FGF ring if every finitely generated left R-module can be embedded in a free left R-module. It is proved that a group ring RG is left FGF...     

Algorithmic Complexity of a Problem of Idempotent Convex Geometry

Properties of the idempotently convex hull of a two-point set in a free semimodule over the idempotent semiring R _{max min} and in a free semimodule over a linearly ordered idempotent semifield are studied. Construction algorithms for this hull are proposed.     

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

Let (S,≤) be an ordered set. Recall that (S,≤) is artinian if every strictly decreasingsequence of elements of S is ?nite, and that (S,≤) is narrow if every subset of pairwiseorder-incomparable elements of S is ?nite. Let S be a commutative monoid. Unl…     

Die absteigende Loewy-Länge von Endomorphismenringen

We give an upper bound for the descending Loewy length of the ring S=End(M), where M is a semi-artinian R-module satisfying some extra-conditions.In case M is of finite length, our main result specializes to the following result, which improves a theorem of Smalø ([3]): The index of nilpotency of Ra(S) is bounded by the number max{l_A¦A is a simple R-module occuring in the socle of M}, where l_A denotes the number of times the isomorphism type of A occurs as a factor in a composition chain of M.     

A mathematical proof of S. Shelah’s theorem on the measure problem and related results

Recently, S. Shelah proved that an inaccessible cardinal is necessary to build a model of set theory in which every set of reals is Lebesgue measurable. We give a simpler and metamathematically free proof of Shelah's result. As a corollary, we get an elementary proof of the following result (without choice axiom): assume there exists an uncountable well ordered set of reals, then there exists a non-measurable set of reals. We also get results about Baire property,K _σ-regularity and Ramsey property.     

Axiomatizability of free <Emphasis Type="Italic">S</Emphasis>-posets

In this work, we investigate the partially ordered monoids S over which the class of free (over a poset) S-posets is axiomatizable. Similar questions for S-sets were considered in papers of V. Gould, S. Bulman-Fleming, and A. A. Stepanova.     

Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis)

For a commutative ring R and a faithfully flat R-algebra S we prove,under mild extra assumptions,that an R-module M is Gorenstein flat if and only if the left S-module S?_R M is Gorenstein flat,and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom R(S,N)is Gorenstein injective.We apply these results to the study of Gorenstein homological dimensions of unbounded complexes.In particular,we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.     

Maximal and Minimal Semilattices on Ordered Sets

The map which takes an element of an ordered set to its principal ideal is a natural embedding of that ordered set into its powerset, a semilattice. If attention is restricted to all finite intersections of the principal ideals of the original ordered set, then an embedding into a much smaller semilattice is obtained. In this paper the question is answered of when this construction is, in a certain arrow-theoretic sense, minimal. Specifically, a characterisation is given, in terms of ideals and filters, of those ordered sets which admit a so-called minimal embedding into a semilattice. Similarly, a candidate maximal semilattice on an ordered set can be constructed from the principal filters of its elements. A characterisation of those ordered sets that extend to a maximal semilattice is given. Finally, the notion of a free semilattice on an ordered set is given, and it is shown that the candidate maximal semilattice in the embedding-theoretic sense is the free object.     

Automorphism group and dimension of ordered sets

It is shown that a finite groupG is isomorphic to the automorphism group of a two-dimensional ordered set if and only if it is a generalized wreath product of symmetric groups over an ordered index set that is a dual tree. Furthermore, every finite abelian group is isomorphic to the full automorphism group of a three-dimensional ordered set. Also every finite group is isomorphic to the automorphism group of an ordered set that does not contain an induced crown with more than four elements.     

CS-indecomposable ordered semigroups

An ordered semigroup S is called CS-indecomposable if the set S × S is the only complete semilattice congruence on S. In the present paper we prove that each ordered semigroup is, uniquely, a complete semilattice of CS-indecomposable semigroups, which means that it can be decomposed into CS-indecomposable components in a unique way. Furthermore, the CS-indecomposable ordered semigroups are exactly the ordered semigroups that do not contain proper filters. Bibliography: 6 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 222–232.     

关于(m,n)-凝聚模与预包络

In this paper,let m,n be two fixed positive integers and M be a right R-module,we define (m,n)-M-flat modules and (m,n)-coherent modules.A right R-module F is called (m,n) M-flat if every homomorphism from an (n,m)-presented right R-module into F factors through a module in addM.A left S-module M is called an (m,n)-coherent module if MR is finitely presented,and for any (n,m)-presented right R-module K,Horn(K,M) is a finitely generated left S-module,where S = End(MR).We mainly characterize (m,n)-coherent modules in terms of preenvelopes (which are monomorphism or epimorphism) of modules.Some properties of (m,n)-coherent rings and coherent rings are obtained as corollaries.