**Contemporary Mathematics**

Volume: 634;
2015;
265 pp;
Softcover

MSC: Primary 16; 17; 94;

Print ISBN: 978-1-4704-1032-2

Product Code: CONM/634

List Price: $105.00

AMS Member Price: $84.00

MAA member Price: $94.50

**Electronic ISBN: 978-1-4704-2264-6
Product Code: CONM/634.E**

List Price: $105.00

AMS Member Price: $84.00

MAA member Price: $94.50

# Noncommutative Rings and Their Applications

Share this page *Edited by *
*Steven Dougherty; Alberto Facchini; André Leroy; Edmund Puczyłowski; Patrick Solé*

This volume contains the Proceedings of an
International Conference on Noncommutative Rings and Their
Applications, held July 1–4, 2013, at the Université
d'Artois, Lens, France. It presents recent developments in the
theories of noncommutative rings and modules over such rings as well
as applications of these to coding theory, enveloping algebras, and
Leavitt path algebras.

Material from the course “Foundations of Algebraic Coding
Theory“, given by Steven Dougherty, is included and provides the
reader with the history and background of coding theory as well as the
interplay between coding theory and algebra. In module theory, many
new results related to (almost) injective modules, injective hulls and
automorphism-invariant modules are presented. Broad generalizations of
classical projective covers are studied and category theory is used to
describe the structure of some modules. In some papers related to
more classical ring theory such as quasi duo rings or clean elements,
new points of view on classical conjectures and standard open problems
are given. Descriptions of codes over local commutative Frobenius
rings are discussed, and a list of open problems in coding theory is
presented within their context.

#### Readership

Graduate students and research mathematicians interested in noncommutative rings and coding theory.

# Table of Contents

## Noncommutative Rings and Their Applications

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Leavitt Path Algebras of Cayley Graphs Arising from Cyclic Groups 112 free
- Characterizations of Almost Injective Modules 1122
- Automorphism-Invariant Modules 1930
- 1. Introduction 1930
- 2. When is an automorphism-invariant module quasi-injective 2031
- 3. Endomorphism rings and structure of automorphism- invariant modules 2233
- 4. Properties of automorphism-invariant modules 2334
- 5. Automorphism-invariant Leavitt path algebras 2536
- 6. Modules invariant under automorphisms of their pure-injective envelope 2637
- References 2839

- Distributive Algebras, Isoclinism, and Invariant Probabilities 3142
- Ideals of the Enveloping Algebra 𝑈(𝑠𝑙₃) 5364
- On Second Submodules 6778
- Open Problems in Coding Theory 7990
- 1. Introduction 7990
- 2. Definitions and Notations 8091
- 3. Fundamental Problem of Coding Theory 8091
- 4. Duality for non-Abelian groups 8192
- 5. Designs and Codes 8394
- 6. MDS Codes 8394
- 7. Doubly-even binary codes 8596
- 8. Codes and Lattices 89100
- 9. Self-Dual Codes 91102
- 10. Decoding Algorithms 92103
- 11. Bounds on Codes 92103
- 12. Covering Radius 93104
- 13. Cyclic Codes 94105
- 14. Optimal 2-error Correcting Codes 94105
- 15. Virtually Self-Dual Weight Enumerator 95106
- References 96107

- Foundations of Algebraic Coding Theory 101112
- 1. Introduction 101112
- 2. Classical Coding Theory 103114
- 3. Connections of Coding Theory in Mathematics 108119
- 4. The Gray Map and a Shock to Classical Coding Theory 110121
- 5. Cyclic Codes over ℤ_{𝕞} 111122
- 6. Frobenius Rings 115126
- 7. Chinese Remainder Theorem 118129
- 8. Generators of codes 120131
- 9. Self-Dual Codes 122133
- 10. Generalizations for Hamming Weight 131142
- References 133144

- Counting ℤ₂ℤ₄-Additive Codes 137148
- From Endomorphism Rings to Some Noteworthy Ideals in Categories of Modules 149160
- 1. Semilocal Rings 150161
- 2. Rings and modules of finite type 150161
- 3. Local functors and Jacobson radical 153164
- 4. The canonical functor \Cal𝐴→\Cal𝐴/\Cal𝐼₁×…×\Cal𝐴/\Cal𝐼_{𝑛} 154165
- 5. Semilocal Categories 155166
- 6. Examples and applications 156167
- 7. Birkhoff’s Theorem for skeletally small preadditive categories and \Mod𝑅 158169
- References 160171

- Lie and Jordan Properties in Group Algebras 163174
- Rad-Projective 𝛿-Covers 175186
- On Lattices of Annihilators 189200
- Clean Elements in Polynomial Rings 197208
- On the Semiprime Smash Product Question 205216
- Minimal Prime Ideals and Derivations 223234
- On Codes over Local Frobenius Non-Chain Rings of Order 16 227238
- On Quasi-Duo Rings 243254
- Isomorphism Invariants of Enveloping Algebras 253264
- Back Cover Back Cover1280