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fuck-final.md

Ring

Chain of class inclusions

Rings ⊃ Commutative rings ⊃ ID ⊃ UFD ⊃ PID ⊃ ED ⊃ Field

Prime

if a | bc then a | b or a | c, i.e. a can't be torn apart if a is prime

Irreducible

if a | bc then either b or c is a unit.

In ID

Prime implies irreducible.

In UFD

Prime implies irreducible, irreducible implies prime.

Euclidean norm

  • n : R/{0} -> Nat
  • n(s) < n(s)n(r)
  • n(u) = 1 iff u is a unit

Multiplicative norm

  • n : R/{0} -> Nat
  • n(ab) = n(a)n(b)
  • n(a) = 0 iff a = 0
  • n(1) = 1
  • if u is a unit then n(u) = 1 or n(u) = -1
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