Math 2.0

Subjects

Associative rings and algebras

449 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 244 of its 245 ideas have a page so far.

Start here

Map of the subject

Basic notions 24 of 25

  1. Ring definition
  2. Noncommutative ring concept
  3. Associative algebra definition
  4. Subring definition
  5. Opposite ring definition
  6. Ring homomorphism definition
  7. Left ideal definition
  8. Two-sided ideal definition
  9. Quotient ring definition
  10. Center of a ring definition
  11. Centraliser definition
  12. Unit of a ring definition
  13. Zero divisor definition
  14. Domain definition
  15. Division ring definition
  16. Idempotent element definition
  17. Complete set of orthogonal idempotents definition
  18. Peirce decomposition statement
  19. Nilpotent element definition
  20. Nil ideal definition
  21. Nilpotent ideal definition
  22. matrix ring definition
  23. Endomorphism ring definition
  24. Characteristic of a ring definition
  25. Rng definition

Modules and ideals 23 of 23

  1. Module over a ring definition
  2. Bimodule definition
  3. Simple module definition
  4. Schur's lemma statement
  5. Endomorphism ring of a simple module is a division ring statement
  6. Semisimple module definition
  7. Characterizations of semisimple modules statement
  8. Indecomposable module definition
  9. Krull–Schmidt theorem statement
  10. Composition series definition
  11. Jordan–Hölder theorem statement
  12. Length of a module definition
  13. Fitting's lemma statement
  14. Faithful module definition
  15. Uniform module definition
  16. Essential submodule definition
  17. Socle of a module definition
  18. Radical of a module definition
  19. Free module definition
  20. Invariant basis number definition
  21. Stably finite ring definition
  22. Dedekind-finite ring definition
  23. Eilenberg swindle concept

Semisimple rings and Wedderburn theory 8 of 8

  1. Simple ring definition
  2. Matrix rings over simple rings are simple statement
  3. Semisimple ring definition
  4. Artin–Wedderburn theorem statement
  5. Wedderburn's theorem on simple Artinian rings statement
  6. Wedderburn's little theorem statement
  7. Maschke's theorem statement
  8. Separable algebra definition

Division rings and central simple algebras 19 of 19

  1. The quaternions object
  2. Quaternion algebra definition
  3. Frobenius theorem on real division algebras statement
  4. Central simple algebra definition
  5. Skolem–Noether theorem statement
  6. Double centralizer theorem statement
  7. Splitting field of a central simple algebra definition
  8. Maximal subfield definition
  9. Degree of a central simple algebra definition
  10. Brauer group definition
  11. Period and index concept
  12. Crossed product algebra definition
  13. Cyclic algebra definition
  14. Albert–Brauer–Hasse–Noether theorem statement
  15. Amitsur's noncrossed product counterexample
  16. Cartan–Brauer–Hua theorem statement
  17. Hua's identity statement
  18. Azumaya algebra definition
  19. Clifford algebra definition

Radicals 10 of 10

  1. Jacobson radical definition
  2. Jacobson radical of an Artinian ring is nilpotent statement
  3. Nakayama's lemma statement
  4. Prime radical definition
  5. Upper nilradical definition
  6. Levitzki radical definition
  7. Brown–McCoy radical definition
  8. Kurosh–Amitsur radical theory concept
  9. Köthe conjecture conjecture
  10. Amitsur's theorem on the radical of polynomial rings statement

Primitive and prime rings 9 of 9

  1. Primitive ring definition
  2. Primitive ideal definition
  3. Jacobson density theorem statement
  4. Prime ring definition
  5. Prime ideal in a noncommutative ring definition
  6. Semiprime ring definition
  7. Semiprimitive ring definition
  8. Reduced ring definition
  9. Kaplansky's theorem on projective modules statement

Local rings and generalizations 9 of 9

  1. Local ring definition
  2. Semilocal ring definition
  3. Semiperfect ring definition
  4. Perfect ring definition
  5. Semiprimary ring definition
  6. Bass's theorem P statement
  7. Lifting idempotents concept
  8. Uniserial module definition
  9. Serial ring definition

Chain, growth and finiteness conditions 19 of 19

  1. Noetherian ring definition
  2. Artinian ring definition
  3. Noetherian module definition
  4. Artinian module definition
  5. Hopkins–Levitzki theorem statement
  6. Artinian reduced rings are semisimple statement
  7. Ore condition definition
  8. Ore localization definition
  9. Goldie's theorem statement
  10. Goldie ring definition
  11. Classical ring of quotients definition
  12. Gelfand–Kirillov dimension definition
  13. Growth of algebras concept
  14. Bergman's gap theorem statement
  15. Kurosh problem problem
  16. Golod–Shafarevich theorem statement
  17. Levitzki's theorem statement
  18. Krull dimension of a noncommutative ring definition
  19. Lenagan's theorem statement

Polynomial identity rings 6 of 6

  1. Polynomial identity ring definition
  2. Amitsur–Levitzki theorem statement
  3. Kaplansky's theorem on PI rings statement
  4. Posner's theorem statement
  5. Central polynomial definition
  6. Generic matrix algebra definition

Constructions 25 of 25

  1. Polynomial ring definition
  2. Free algebra definition
  3. Tensor algebra definition
  4. Exterior algebra definition
  5. Tensor product of algebras definition
  6. Group ring definition
  7. Monoid ring definition
  8. Kaplansky's conjectures conjecture
  9. Kaplansky's zero divisor conjecture conjecture
  10. Skew polynomial ring definition
  11. Ore extension definition
  12. Weyl algebra definition
  13. Dixmier conjecture conjecture
  14. Ring of differential operators definition
  15. Formal power series definition
  16. Laurent polynomial ring definition
  17. Skew group ring definition
  18. Smash product definition
  19. Path algebra definition
  20. Incidence algebra definition
  21. Universal enveloping algebra definition
  22. Poincaré–Birkhoff–Witt theorem statement
  23. Leavitt path algebra definition
  24. Cohn localization concept
  25. Direct limit of rings definition

Morita theory 5 of 5

  1. Morita equivalence definition
  2. Morita equivalence of a ring with its matrix rings statement
  3. Progenerator definition
  4. Morita duality concept
  5. Morita invariant concept

Homological methods 25 of 25

  1. Projective module definition
  2. Injective module definition
  3. Baer's criterion statement
  4. Flat module definition
  5. Injective hull definition
  6. Projective cover definition
  7. Ext functor definition
  8. Tor functor definition
  9. Global dimension definition
  10. Weak dimension definition
  11. Hereditary ring definition
  12. Semihereditary ring definition
  13. Frobenius algebra definition
  14. Quasi-Frobenius ring definition
  15. Self-injective ring definition
  16. Hochschild homology definition
  17. Cyclic homology definition
  18. Koszul algebra definition
  19. Koszul duality concept
  20. Gerstenhaber algebra definition
  21. Deformation of algebras concept
  22. Deformation quantization concept
  23. Kontsevich formality theorem statement
  24. Algebraic K-theory concept
  25. Grothendieck group definition

Representations of algebras 18 of 18

  1. Quiver definition
  2. Representation of a quiver definition
  3. Gabriel's theorem statement
  4. Dynkin quiver definition
  5. Kac's theorem statement
  6. Auslander–Reiten theory concept
  7. Auslander–Reiten quiver definition
  8. Almost split sequence definition
  9. Tilting theory concept
  10. Bernstein–Gelfand–Ponomarev reflection functors concept
  11. Cluster algebra definition
  12. Cartan matrix of an algebra definition
  13. Nakayama algebra definition
  14. Cellular algebra definition
  15. Hecke algebra definition
  16. Temperley–Lieb algebra definition
  17. Group representation concept
  18. Modular representation theory concept

Orders 6 of 6

  1. Order (ring theory) definition
  2. Maximal order definition
  3. Eichler order definition
  4. Hurwitz quaternion order example
  5. Hurwitz quaternions object
  6. Jordan–Zassenhaus theorem statement

Hopf algebras and quantum groups 18 of 18

  1. Coalgebra definition
  2. Bialgebra definition
  3. Hopf algebra definition
  4. Antipode definition
  5. Comodule definition
  6. Sweedler notation concept
  7. Hopf module definition
  8. Integral of a Hopf algebra definition
  9. Larson–Sweedler theorem statement
  10. Nichols–Zoeller theorem statement
  11. Cartier–Milnor–Moore theorem statement
  12. Quasitriangular Hopf algebra definition
  13. Quantum group definition
  14. Quantum enveloping algebra definition
  15. Yang–Baxter equation definition
  16. Ribbon Hopf algebra definition
  17. Taft algebra example
  18. Nichols algebra definition

Rings with additional structure 9 of 9

  1. Graded ring definition
  2. Filtered algebra definition
  3. Superalgebra definition
  4. Differential graded algebra definition
  5. Ring with involution definition
  6. Derivation definition
  7. Banach algebra definition
  8. Topological ring definition
  9. Jacobson topology definition

Conditions on elements 5 of 5

  1. von Neumann regular ring definition
  2. Unit-regular ring definition
  3. Clean ring definition
  4. Boolean ring definition
  5. Jacobson's conjecture conjecture

Generalizations of rings 5 of 5

  1. Semiring definition
  2. Near-ring definition
  3. Nonassociative ring definition
  4. Jordan algebra definition
  5. Octonions object

Computation 1 of 1

  1. Noncommutative Gröbner basis concept

More ideas in this subject

Concepts 23

Theorems and results 14

Examples 5

Objects 3

Definitions 187

This subject is MSC class 16. See it on zbMATH Open.