Editing Principal ideal domain (section)

=== Non-examples ===
Examples of integral domains that are not PIDs:

* <math>\mathbb{Z}[\sqrt{-3}]</math> is an example of a ring that is not a [[unique factorization domain]], since <math>4 = 2\cdot 2 = (1+\sqrt{-3})(1-\sqrt{-3}).</math> Hence it is not a principal ideal domain because principal ideal domains are unique factorization domains. Also, <math>\langle 2, 1+\sqrt{-3} \rangle</math> is an ideal that cannot be generated by a single element.
* <math>\mathbb{Z}[x]</math>: the ring of all polynomials with integer coefficients. It is not principal because <math>\langle 2, x \rangle</math> is  an ideal that cannot be generated by a single polynomial.
* <math>K[x, y, \ldots],</math>  the [[Polynomial_ring#Definition (multivariate case)|ring of polynomials in at least two variables]] over a ring {{mvar|K}} is not principal, since the ideal <math>\langle x, y \rangle</math> is not principal.
* Most [[ring of algebraic integers|rings of algebraic integers]] are not principal ideal domains. This is one of the main motivations behind Dedekind's definition of [[Dedekind domain]]s, which allows replacing [[unique factorization]] of elements with unique factorization of ideals. In particular, many <math>\mathbb{Z}[\zeta_p],</math> for the [[Root of unity|primitive p-th root of unity]] <math>\zeta_p,</math> are not principal ideal domains.<ref>{{Cite web|first = James|last=Milne|authorlink = James Milne (mathematician)|title=Algebraic Number Theory|url=https://www.jmilne.org/math/CourseNotes/ANT301.pdf|pages=5}}</ref> The [[Class number (number theory)|class number]] of a ring of algebraic integers gives a measure of "how far away" the ring is from being a principal ideal domain.