二次李三系的分解定理

有限维李三系称为二次的,如果它容许一个非退化的不变对称双线性型.无论是李三系分解为不可分解理想的直和,还是二次李三系分解为不可分解非退化理想的正交直和,证明了这两类分解在同构意义下都是唯一的.     

Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

Jordan李超代数的分解再论

代数A称为不可分解的,如果A不能分解成理想的直和.满足C(Lo)=C(L)={0}的Jordan李超代数L能够分解成不可分解理想的直和,这种分解在不计理想次序的前提下是唯一的.并证明了完备Jordan李超代数的一些性质.     

Graded level zero integrable representations of affine Lie algebras

We study the structure of the category of integrable level zero representations with finite dimensional weight spaces of affine Lie algebras. We show that this category possesses a weaker version of the finite length property, namely that an indecomposable object has finitely many simple constituents which are non-trivial as modules over the corresponding loop algebra. Moreover, any object in this category is a direct sum of indecomposables only finitely many of which are non-trivial. We obtain a parametrization of blocks in this category.

     

A note on the structure of injective diagrams

We determine all indecomposable injective diagrams over a poset I with values in some category R-Mod of modules and characterize the case when every injective diagram decomposes into a direct sum of indecomposables. Moreover, we show that injective diagrams have a standard form dual to that of projective diagrams iff I is a noetherian poset.     

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R, it is shown that the local duality establishes a bijection between the preinjective left R-modules and the preprojective right R-modules, and any preinjective left R-module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R-modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361-376] for hereditary rings.     

A Krull-Schmidt theorem for one-dimensional rings of finite Cohen-Macaulay type

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.     

Lifting Modules with Indecomposable Decompositions

A module M is called a “lifting module” if, any submodule A of M contains a direct summand B of M such that A/B is small in M/B. This is a generalization of projective modules over perfect rings as well as the dual of extending modules. It is well known that an extending module with ascending chain condition (a.c.c.) on the annihilators of its elements is a direct sum of indecomposable modules. If and when a lifting module has such a decomposition is not known in general. In this article, among other results, we prove that a lifting module M is a direct sum of indecomposable modules if (i) rad(M ^(I)) is small in M ^(I) for every index set I, or, (ii) M has a.c.c. on the annihilators of (certain) elements, and rad(M) is small in M.     

On Modules with Finite Exchange Property

The purpose of this paper is to answer an open problem proposed by Matlis on modules with the finite exchange property. The problem is: Let {M _i | i ∈ I} be an indexed family of indecomposable injective modules and M a direct sum of M _i , i.e. M = ⊕{M _i | i ∈ I} and N a summand of M. Is N a direct sum of indecomposable injective modules, i.e. N = ⊕{M _j |j ∈ J, J ? I}? The answer to this problem is affirmative for module M with finite exchange property.     

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.     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

Dedekind整环上的不可分解扭模

We give a direct proof to the classification of all indecomposable torsion modules over a Dedekind ring. As an application, we classify all indecomposable locally-nilpotent modules over the polynomial algebra with one variable over a field.     

Jordan李超代数的一些性质

代数A称为不可分解的,如果A不能分解成理想的直和.证明了满足C(L_o)=C(L)={0}的Jordan李超代数L一些重要性质.     

CS Modules Relative to a Torsion Theory

In this article we introduce two new concepts, those of a τ-CS and a s-τ-CS module, which are both torsion-theoretic analogues of CS modules. We investigate their relationship with the familiar concepts of τ-injective, τ-simple and τ-uniform modules and compare them with τ-complemented (τ-injective) modules, which were considered by other authors as torsion-theoretic analogues of CS modules. We are interested in decomposing a relatively CS module into indecomposable submodules, and in determining when a direct sum of relatively CS modules is relatively CS. This paper forms part of the Ph.D. thesis of the first author, written under the supervision of the second author. The first author gratefully acknowledges the support of the Commonwealth Scholarship and Fellowship Committee of New Zealand.     

An indecomposable direct summand of a serial module which is not uniserial

Abstract

We present a construction of indecomposable direct summands of serial modules which are not uniserial.     

Conjugacy of idempotents in jordan pairs

The injectives in the category of associative rings and homomorphisms with accessible images are investigated.It is shown that every such ring has an identity and is a direct sum of finitely many indecomposable injectives.The indecomposable injectives are shown to be simple when they are algebras over fields other than Z_p and in the remaining characteristic p case to have only three ideals.A necessary and sufficient condition is obtained for certain finite direct sums of indecomposable injectives to be injective.