Principle Ideal Domains
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A Principal Ideal Domain (PID) is a domain, \(R\) with the property that every ideal is principal, i.e., for each ideal \(I\), we have \(I = (a) \) for some \(a \in R\).
All PIDs are EDs, but the opposite is not true. For example the ring \(\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]=\{a+b\frac{1+\sqrt{-19}}{2} | a,b \in \mathbb{Z}\} \)
If \(R\) is a PID with \(a,b\in \R\), then:
- \((a,b)=(g)\) for some \(g\in R \) and any such \(g\) is a gcd of \(a\) and \(b\).
- The gcd of \(a\) and \(b\) is unique up to multiplication by a unit
If R is not only a PID but a Euclidean domain with norm function \(N\), then the Euclidean algorithm can be used to compute a gcd of any two nonzero \(a, b \in R\).
In a PID, irreducible elements are prime.
PID implies Notherian Rings.
