Algorithms for finite rings

Leiden Repository

Algorithms for finite rings

Type: Doctoral Thesis
Title: Algorithms for finite rings
Author: Ciocanea Teodorescu I.
Issue Date: 2016-06-22
Keywords: Deterministic polynomial time algorithm
Finite rings
Modules
Jacobson radical
Separability
Abstract: In this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules that are finite. The first main result of this thesis is a solution to the module isomorphism problem in the finite case. Further, we show how to compute a set of generators of minimal cardinality for a given finite module, and how to construct projective covers and injective hulls. We also describe tests for module simplicity, projectivity, and injectivity, and constructive tests for existence of surjective module homomorphisms between two finite modules, one of which is projective. As a negative result, we show that the problem of testing for existence of injective module homomorphisms between two finite modules, one of which is projective, is NP-complete. The last part of the thesis is concerned with finding a good working approximation of the Jacobson radical of a finite ring, that is, a two-sided nilpotent ideal such that the corresponding quotient ring is “almost” semisimple. The notion we use to approximate semisimplicity is that of separability.
Promotor: Supervisor: H.W. Lenstra, K. Belabas
Faculty: Science
University: Leiden University and l'Université de Bordeaux
Handle: http://hdl.handle.net/1887/40676
 

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application/pdf Introduction 256.9Kb View/Open
application/pdf List of Symbols 204.1Kb View/Open
application/pdf Chapter 1 377.1Kb View/Open
application/pdf Chapter 2 404.9Kb View/Open
application/pdf Chapter 3 289.1Kb View/Open
application/pdf Chapter 4 350.4Kb View/Open
application/pdf Chapter 5 324.0Kb View/Open
application/pdf Chapter 6 547.7Kb View/Open
application/pdf Bibliography_Index 166.7Kb View/Open
application/pdf Abstract 153.4Kb View/Open
application/pdf Summary in French 153.7Kb View/Open
application/pdf Summary in Dutch 153.5Kb View/Open
application/pdf Acknowledgements_Curriculum Vitae 122.7Kb View/Open
application/pdf Propositions 138.5Kb View/Open

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