Every ideal of Z is a principal ideal

Dependencies:

  1. Principal ideal
  2. Integer Division Theorem

Let I be an ideal of Z. Then I is a principal ideal, i.e. I is of the form nZ for some nZ.

Proof

Let n be the smallest positive element of I. Let aI.

By the integer division theorem, a=qn+r, where 0r<n. r=aqnI by absorption and commutativity. Since rI, r<n and n is the smallest positive element of I, r=0.

Therefore, every element of I is a multiple of n. Also, all multiples of n are in I by the absorption property.

Therefore, I=nZ.

Dependency for: None

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Ideal
  4. Identity of a group is unique
  5. Subgroup
  6. Inverse of a group element is unique
  7. Conditions for a subset to be a subgroup
  8. Condition for a subset to be a subgroup
  9. Conditions for a subset of a ring to be a subring
  10. Principal ideal
  11. Integer Division Theorem