Ring

Integral domain

An integral domain is a nonzero commutative ring with no nonzero zero-divisors, #m/def/ring i.e. iff or . This gives rise the the cancellation property, since all nonzero elements are epic and monic: and implies .

Proof

Since and , it follows and hence .

Note that by moving to the Field of fractions we can get cancellation in the normal way.

Properties

  1. A finite integral domain is a field
  2. The characteristic of an integral domain is 0 or prime
  3. for commutative is an integral domain iff is prime
  4. is an integral domain iff is an integral domain
  5. All primes are irreducible in an integral domain

Other results

See also

#state/tidy | #lang/en | #SemBr