Principal Ideal Domain - Properties

Properties

In a principal ideal domain, any two elements a,b have a greatest common divisor, which may be obtained as a generator of the ideal (a,b).

All Euclidean domains are principal ideal domains, but the converse is not true. An example of a principal ideal domain that is not a Euclidean domain is the ring In this domain no q and r exist, with 0≤|r|<4, so that, despite and 4 having a greatest common divisor of 2.

Every principal ideal domain is a unique factorization domain (UFD). The converse does not hold since for any field K, K is a UFD but is not a PID (to prove this look at the ideal generated by It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element).

1. Every principal ideal domain is Noetherian.
2. In all unital rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
3. All principal ideal domains are integrally closed.

The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.

Let A be an integral domain. Then the following are equivalent.

1. A is a PID.
2. Every prime ideal of A is principal.
3. A is a Dedekind domain that is a UFD.
4. Every finitely generated ideal of A is principal (i.e., A is a Bézout domain) and A satisfies the ascending chain condition on principal ideals.
5. A admits a Dedekind–Hasse norm.

A field norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID. (4) compares to:

• An integral domain is a UFD if and only if it is a GCD domain (i.e., a domain where every two elements has a greatest common divisor) satisfying the ascending chain condition on principal ideals.

An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two. A Bézout domain is thus a GCD domain, and (4) gives yet another proof that a PID is a UFD.