Metamath Proof Explorer

Table of Contents - 10. BASIC ALGEBRAIC STRUCTURES

  1. Monoids
    1. Magmas
    2. Identity elements
    3. Iterated sums in a magma
    4. Magma homomorphisms and submagmas
    5. Semigroups
    6. Definition and basic properties of monoids
    7. Monoid homomorphisms and submonoids
    8. Iterated sums in a monoid
    9. Free monoids
    10. Examples and counterexamples for magmas, semigroups and monoids
  2. Groups
    1. Definition and basic properties
    2. Group multiple operation
    3. Subgroups and Quotient groups
    4. Cyclic monoids and groups
    5. Elementary theory of group homomorphisms
    6. Isomorphisms of groups
    7. Group actions
    8. Centralizers and centers
    9. The opposite group
    10. Symmetric groups
    11. p-Groups and Sylow groups; Sylow's theorems
    12. Direct products
    13. Free groups
    14. Abelian groups
    15. Simple groups
  3. Rings
    1. Multiplicative Group
    2. Non-unital rings ("rngs")
    3. Ring unity (multiplicative identity)
    4. Semirings
    5. Unital rings
    6. Opposite ring
    7. Divisibility
    8. Ring primes
    9. Homomorphisms of non-unital rings
    10. Ring homomorphisms
    11. Nonzero rings and zero rings
    12. Local rings
    13. Subrings
    14. Categories of rings
  4. Division rings and fields
    1. Definition and basic properties
    2. Sub-division rings
    3. Absolute value (abstract algebra)
    4. Star rings
  5. Left modules
    1. Definition and basic properties
    2. Subspaces and spans in a left module
    3. Homomorphisms and isomorphisms of left modules
    4. Subspace sum; bases for a left module
  6. Vector spaces
    1. Definition and basic properties
  7. Subring algebras and ideals
    1. Subring algebras
    2. Left ideals and spans
    3. Two-sided ideals and quotient rings
    4. Principal ideal rings. Divisibility in the integers
    5. Left regular elements. More kinds of rings
  8. The complex numbers as an algebraic extensible structure
    1. Definition and basic properties
    2. Ring of integers
    3. Algebraic constructions based on the complex numbers
    4. Signs as subgroup of the complex numbers
    5. Embedding of permutation signs into a ring
    6. The ordered field of real numbers
  9. Generalized pre-Hilbert and Hilbert spaces
    1. Definition and basic properties
    2. Orthocomplements and closed subspaces
    3. Orthogonal projection and orthonormal bases