Principle Ideal Domains

Principal Ideal Domain

A integral domain is called a principal ideal domain if every ideal is principle.

e.g.

• Every field is a principal ideal domain. In fact, let be an ideal, if is non-trivial, then it contains a non-zero element . Since is a field, is a unit, and thus . So any field only has two trivial ideals: and ;
• Both and the ring of polynomials over a field are PIDs;
• for some field is not a principal ideal domain. The ideal is not principle;
• In , is not principle.
  

Proposition

is a principal ideal domain.

Proof Suppose is a non trivial ideal. There exists at least one positive integer in it. Pick the smallest positive integer . We claim . For any , by the remainder theorem, we have for some with . Since , it follows that . As is the smallest positive integer in , we must have . Therefore .

Remark

Indeed, any Euclidean domain is a principle ideal domain. (see proposition.)

Proposition

is a principal ideal domain for any field .

Proof

Euclidean Algorithm

The gcd of polynomials , and the linear combination that gives it can be found by the following algorithm. Compute Then up to a unit is the gcd and backward substitution gives the linear combination.

Proposition

In an integral domain every prime is irreducible.

Proof Suppose is a prime with . Then implies or . Without loss of generality, suppose . By the equation, as well, so and are associates, thus is a unit.

Nilpotent & Idempotent

Let be a ring. is nilpotent if there is such that . is idempotent if .

Proposition

If is nilpotent then is invertible.

Recall the fundamental theorem of arithmetic:

Every integer greater than can either be prime or represented uniquely as a product of prime numbers.

Link to original

Similarly for polynomials:

Theorem

Let be a field. Every non-constant polynomial in can be written uniquely (up to reordering) as for a unit and , monic irreducible and .

Both theorems are in fact one theorem:

Unique Factorization in A PID

Every principle ideal domain is a unique factorization domain.