Metamath Proof Explorer

Table of Contents - 10. BASIC ALGEBRAIC STRUCTURES

  1. Monoids
    1. Magmas
    2. Identity elements
    3. Iterated sums in a magma
    4. Semigroups
    5. Definition and basic properties of monoids
    6. Monoid homomorphisms and submonoids
    7. Iterated sums in a monoid
    8. Free monoids
    9. 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. Ring unit
    3. Definition and basic properties of unital rings
    4. Opposite ring
    5. Divisibility
    6. Ring primes
    7. Ring homomorphisms
  4. Division rings and fields
    1. Definition and basic properties
    2. Subrings of a ring
    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. Ideals
    1. The subring algebra; ideals
    2. Two-sided ideals and quotient rings
    3. Principal ideal rings. Divisibility in the integers
    4. Nonzero rings and zero rings
    5. Left regular elements. More kinds of rings
  8. Associative algebras
    1. Definition and basic properties
  9. Abstract multivariate polynomials
    1. Definition and basic properties
    2. Polynomial evaluation
    3. Additional definitions for (multivariate) polynomials
    4. Univariate polynomials
    5. Univariate polynomial evaluation
  10. 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
  11. Generalized pre-Hilbert and Hilbert spaces
    1. Definition and basic properties
    2. Orthocomplements and closed subspaces
    3. Orthogonal projection and orthonormal bases