Principal ideal domain

In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).

Examples are the ring of integers, all fields, and rings of polynomials in one variable with coefficients in a field. All euclidean domains are principal ideal domains, but the converse is not true.

An example of a non PID is the ring Z[X] of all polynomials with integer coefficients. It is not principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.

Properties

In a principal ideal domain, any two elements have a greatest common divisor, and almost always have more than one.

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

Every principal ideal domain is Noetherian.
In all rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
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.

So that PID $\subseteq$ Dedekind $\cap$ UFD . However there is another theorem which states that any unique factorisation domain that is a Dedekind domain is also a principal ideal domain. Thus we get the reverse inclusion Dedekind $\cap$ UFD $\subseteq$ PID, but then this shows equality and hence, Dedekind $\cap$ UFD = PID.

An example of a principal ideal domain that is not a euclidean domain is the ring $\mathbf{Z}\left[\frac{1+\sqrt{-19}}{2}\right]$ (Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34-38, 1973).

Categories: Commutative algebra