Noncommutative Rings and Their Applications.

Cover -- Title page -- Contents -- Preface -- Leavitt Path Algebras of Cayley Graphs Arising from Cyclic Groups -- References -- Characterizations of Almost Injective Modules -- 1. Introduction and Definitions -- 2. Main Result -- 3. Applications -- 4. Almost self-injectivity -- References -- Automorphism-Invariant Modules -- 1. Introduction -- 2. When is an automorphism-invariant module quasi-injective -- 3. Endomorphism rings and structure of automorphism- invariant modules -- 4. Properties of automorphism-invariant modules -- 5. Automorphism-invariant Leavitt path algebras -- 6. Modules invariant under automorphisms of their pure-injective envelope -- References -- Distributive Algebras, Isoclinism, and Invariant Probabilities -- 1. Introduction -- 2. Preliminaries -- 3. Distributive algebras -- 4. Isoclinism and isologism -- 5. Applications -- References -- Ideals of the Enveloping Algebra ( ₃) -- 1. Introduction -- 2. Motivation -- 3. The enveloping algebra as an adjoint module -- 4. A generalized Weyl algebra -- 5. Main results -- 6. Computational results -- 7. The remaining proofs -- 8. Example -- References -- On Second Submodules -- 1. Introduction -- 2. Second Submodules and Attached Primes -- 3. Second Radical Of Modules -- References -- Open Problems in Coding Theory -- 1. Introduction -- 2. Definitions and Notations -- 3. Fundamental Problem of Coding Theory -- 4. Duality for non-Abelian groups -- 5. Designs and Codes -- 6. MDS Codes -- 7. Doubly-even binary codes -- 8. Codes and Lattices -- 9. Self-Dual Codes -- 10. Decoding Algorithms -- 11. Bounds on Codes -- 12. Covering Radius -- 13. Cyclic Codes -- 14. Optimal 2-error Correcting Codes -- 15. Virtually Self-Dual Weight Enumerator -- References -- Foundations of Algebraic Coding Theory -- 1. Introduction -- 2. Classical Coding Theory.

3. Connections of Coding Theory in Mathematics -- 4. The Gray Map and a Shock to Classical Coding Theory -- 5. Cyclic Codes over ℤ_{ } -- 6. Frobenius Rings -- 7. Chinese Remainder Theorem -- 8. Generators of codes -- 9. Self-Dual Codes -- 10. Generalizations for Hamming Weight -- References -- Counting ℤ₂ℤ₄-Additive Codes -- 1. Introduction -- 2. Free codes -- 3. Additive Codes -- References -- From Endomorphism Rings to Some Noteworthy Ideals in Categories of Modules -- 1. Semilocal Rings -- 2. Rings and modules of finite type -- 3. Local functors and Jacobson radical -- 4. The canonical functor \Cal →\Cal /\Cal ₁×…×\Cal /\Cal _{ } -- 5. Semilocal Categories -- 6. Examples and applications -- 7. Birkhoff's Theorem for skeletally small preadditive categories and \Mod -- References -- Lie and Jordan Properties in Group Algebras -- 1. Basic Definitions -- 2. Lie Nilpotence -- 3. Unique Commutators and the LC Property -- 4. Jordan Nilpotence -- References -- Rad-Projective -Covers -- 1. Preliminaries -- 2. Rad-Projective -Covers -- 3. Rad- -Lifting Modules -- 4. 3- -Covers and - -Covers -- References -- On Lattices of Annihilators -- 1. Introduction -- 2. Around modularity -- 3. Main construction -- 4. Finite dimensional case -- References -- Clean Elements in Polynomial Rings -- Introduction -- 1. 2-primal coefficient ring -- 2. When ( [ ]) forms a subring of [ ] -- References -- On the Semiprime Smash Product Question -- 1. The question -- 2. Conditions on -- 3. Conditions on -- 4. Conclusion -- References -- Minimal Prime Ideals and Derivations -- References -- On Codes over Local Frobenius Non-Chain Rings of Order 16 -- 1. Introduction and Overview -- 2. Preliminaries -- 3. Local rings of order 16 -- 4. Codes Over Local Frobenius Rings of order 16 -- References -- On Quasi-Duo Rings.

1. Introduction and preliminaries -- 2. Some general results -- 3. One-sided ideals of quasi-duo rings -- 4. On quasi-duo skew polynomial rings and ℤ-graded rings -- References -- Isomorphism Invariants of Enveloping Algebras -- 1. Introduction -- 2. Preliminaries -- 3. Fox-type problems -- 4. Powers of the augmentation ideal -- 5. Low dimensional nilpotent Lie algebras -- 6. Positive characteristic and restricted Lie algebras -- 7. Other observations -- 8. Open Problems -- References -- Back Cover.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.