784 research outputs found
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
FINITELY GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
This thesis covers the main theory of modules: modules, submodules, cosets, quotient modules, homomorphisms, ideals, direct sums, and some related topics. Using these notions, a theorem on the structure of finitely generated modules over domains of principal ideals is proved. As an application of this theorem, the theory of the structure of normal forms of matrices over various fields is presented
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
A Lagrangian representation of tangles
We construct a functor from the category of oriented tangles in R^3 to the
category of Hermitian modules and Lagrangian relations over Z[t,t^{-1}]. This
functor extends the Burau representations of the braid groups and its
generalization to string links due to Le Dimet.Comment: 36 pages, 8 figure
Brauer-Thrall for totally reflexive modules
Let R be a commutative noetherian local ring that is not Gorenstein. It is
known that the category of totally reflexive modules over R is representation
infinite, provided that it contains a non-free module. The main goal of this
paper is to understand how complex the category of totally reflexive modules
can be in this situation.
Local rings (R,m) with m^3=0 are commonly regarded as the structurally
simplest rings to admit diverse categorical and homological characteristics.
For such rings we obtain conclusive results about the category of totally
reflexive modules, modeled on the Brauer-Thrall conjectures. Starting from a
non-free cyclic totally reflexive module, we construct a family of
indecomposable totally reflexive R-modules that contains, for every n in N, a
module that is minimally generated by n elements. Moreover, if the residue
field R/m is algebraically closed, then we construct for every n in N an
infinite family of indecomposable and pairwise non-isomorphic totally reflexive
R-modules, that are all minimally generated by n elements. The modules in both
families have periodic minimal free resolutions of period at most 2.Comment: Final version; 34 pp. To appear in J. Algebr
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