Tertiary decompositions in lattice modules

The theory of associated prime ideals of anR-module, and of tertiary decompositions, generalizes toL-modules, whereL is a complete modular lattice and anL-moduleM is a complete modular lattice together with an appropriate module actionp:L×MM. Given appropriate chain conditions onL andM, the theory of associated prime ideals, existence and uniqueness properties for tertiary decompositions, and a form of the Krull intersection theorem all hold in generalized form. If more stringent conditions apply, the theory reduces to a generalized theory of primary decompositions and a second uniqueness theorem holds. The theory can be applied to congruence lattices of algebras in congruence-modular varieties of algebras, using the generalized commutator operation. An important special case is the theory of finite groups, where the descending chain condition allows a natural choice of a distinguished tertiary decomposition and this yields a canonical decomposition of any finite group as a subdirect product of cotertiary finite groups. The group-theoretic application of the tertiary theory yields elementary structure theorems about Galois extensions of fields, where the tertiary decomposition of the Galois group transforms into a representation of a Galois extension as a compositum. For example, given a fieldF, there are distinguished tertiary field extensions ofF, of which all other finite Galois extensions ofF are compositums.Presented by Bjarni Jónsson.     

Direct sum decompositions of infinitely generated modules

Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull-Schmidt Theorem. Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally. In this paper we present an example showing that Green's Indecomposability Theorem fails for infinitely generated modules. By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails.

     

On direct decompositions in modules over group rings

In the theory of infinite groups, one of the most important useful generalizations of the classical Maschke theorem is the Kovačs-Newman theorem, which establishes sufficient conditions for the existence of G-invariant complements in modules over a periodic group G finite over the center. We genralize the Kovačs-Newman theorem to the case of modules over a group ring KG, where K is a Dedekind domain. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 255–261, February, 1997.     

Generators for decompositions of tensor products of modules

If K is a field of finite characteristic p, G is a cyclic group of order q = p ^α , U and W are indecomposable KG-modules, and p ≥ dim U + dim W ? 1, we describe how to find a generator for each of the indecomposable components of the KG-module \({U \otimes W}\).     

Injective modules and linear growth of primary decompositions

The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal in a commutative Noetherian ring has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , there exists a minimal primary decomposition with for all . The generalization involves a finitely generated -module and several ideals; the short proof is based on the theory of injective -modules.

     

Direct decompositions of Artinian modules over hypercyclic groups

Let G be a hypercyclic group and A be an Artianian G-module. A class of simple G-modules is defined and it is proved that there exists a direct decomposition A=C B, where C is a G-submodule, each G-composition factor of which belongs to the class , and B is a G-submodule that does not have G-composition factors belonging to .Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 930–934, July–August, 1991.     

Geometric regularity of direct-sum decompositions in some classes of modules

In this paper, we show that modules with semilocal endomorphism rings appear in abundance in applications, that their direct-sum decompositions are described by the so-called Krull monoids, and that this implies a geometric regularity of the direct-sum decompositions of these modules. Their direct-sum decompositions into indecomposables are not necessarily unique in the sense of the Krull-Schmidt theorem. The application of the theory of Krull monoids to the study of direct-sum decompositions of modules has been developed during the last five years. After a quick survey of the results obtained in this direction, we concentrate in particular on the abundance of examples. At present, these examples are scattered in the literature, and we try to collect them in a systematic way. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 231–244, 2004.     

Minimal decompositions and iterative methods

Summary. The aim of this paper is to prove some Babuška–Brezzi type conditions which are involved in the mortar spectral element discretization of the Stokes problem, for several cases of nonconforming domain decompositions. ID=" <E5>Dedicated to Olof B. Widlund on the occasion of his 60th birthday</E5>     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.     

Matrix equivalence and isomorphism of modules

It is well known that if A and B are n × m matrices over a ring R, then coker A ? coker B does not imply A and B are equivalent. An elementary proof is given that the implication does hold if 1 is in the stable range of R. Furthermore, for certain R (including commutative rings), if A is block diagonal and B is block upper triangular with the same diagonal blocks as A, then coker A ? coker B implies A and B are equivalent under a special equivalence. This extends results of Roth and Gustafson. As a corollary, a theorem on decomposition of modules is obtained.     

Indecomposable decompositions of modular standard modules for two families of association schemes

We investigate the indecomposable decomposition of the modular standard modules of two families of association schemes of finite order. First, we show that, for each prime number p, the standard module over a field F of characteristic p of a residually thin scheme S of p-power order is an indecomposable FS-module. Second, we describe the indecomposable decomposition of the standard module over a field of positive characteristic of a wreath product of finitely many association schemes of rank 2.     

Generators for decompositions of tensor products of modules associated with standard Jordan partitions

If K is a field of finite characteristic p, G is a cyclic group of order q = p^α, U and W are indecomposable KG-modules with dim U = m and dim W = n, and λ(m,n,p) is a standard Jordan partition of mn, we describe how to find a generator for each of the indecomposable components of the KG-module U ? W.     

Spectral division methods for block generalized Schur decompositions

We provide a different perspective of the spectral division methods for block generalized Schur decompositions of matrix pairs. The new approach exposes more algebraic structures of the successive matrix pairs in the spectral division iterations and reveals some potential computational difficulties. We present modified algorithms to reduce the arithmetic cost by nearly 50%, remove inconsistency in spectral subspace extraction from different sides (left and right), and improve the accuracy of subspaces. In application problems that only require a single-sided deflating subspace, our algorithms can be used to obtain a posteriori estimates on the backward accuracy of the computed subspaces with little extra cost.

     

Efficient indexing methods for recursive decompositions of Bayesian networks

We consider efficient indexing methods for conditioning graphs, which are a form of recursive decomposition for Bayesian networks. We compare two well-known methods for indexing, a top-down method and a bottom-up method, and discuss the redundancy that each of these suffer from. We present a new method for indexing that combines the advantages of each model in order to reduce this redundancy. We also introduce the concept of an update manager, which is a node in the conditioning graph that controls when other nodes update their current index. Empirical evaluations over a suite of standard test networks show a considerable reduction both in the amount of indexing computation that takes place, and the overall runtime required by the query algorithm.     

Matrix methods for polynomials

For every complex polynomial a set of special matrices with the characteristic polynomial p will be given. This enables us to apply the Gerschgorin theorem.