ring theory (idea)

There are quite a few related nodes on everything.

• Basic definitions:
  • ring
  • ring homomorphism
  • subring
  • quotient ring
  • centre, centralizer
  • ideal, maximal ideal, prime ideal
  • module, tensor product

• Types of rings:
  • division ring
  • field, subfield, field extension, finite field, field of fractions, characteristic, Frobenius endomorphism
  • integral domain
  • unique factorization domain
  • principal ideal domain
  • Euclidean ring
  • Noetherian

• Examples of rings:
  • Quaternions (see also octonions)
  • Weyl algebra
  • Gaussian integers
  • polynomials, Laurent polynomials
  • rational functions in one variable
  • integers modulo n
  • noncommutative geometry
  • generators and relations for algebras

• Properties of elements in rings
  • multiplicative group,unit
  • associate
  • prime
  • irreducible
  • divides
  • infinite divisor chain

• Theorems about rings
  • Wedderburn's theorem about finite division rings, proof of Wedderburn's theorem about finite division rings
  • proof that the multiplicative group of a finite field is cyclic
  • All finite fields are isomorphic to GF(p^n)
  • invertible matrix theorem
  • Galois theory
  • The Hilbert Basis Theorem
  • Chinese Remainder Theorem