Principal ideal domain
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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 <math>\left\langle X,Y \right\rangle <math>. 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 <math> \subseteq <math> Dedekind <math>\cap<math> 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 <math>\cap<math>UFD <math> \subseteq <math>PID, but then this shows equality and hence, Dedekind <math>\cap<math> UFD = PID.
An example of a principal ideal domain that is not a Euclidean domain is the ring <math>\mathbf{Z}\left[\frac{1+\sqrt{-19}}{2}\right]<math> (Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34-38, 1973).de:Hauptidealring fr:Anneau principal