example of a Bezout domain that is not a PID

Let 𝔸 be the ring of all algebraic numbersMathworldPlanetmath whose minimal polynomials are in [x]; i.e. (http://planetmath.org/Ie), every element of 𝔸 is an algebraic integerMathworldPlanetmath.

In the following example, ideals are considered to be of 𝔸 unless indicated otherwise via intersection with a subring of 𝔸.

Let I be a ideal of 𝔸. Then there exists a positive integer n and α1,,αn𝔸 with I=α1,,αn. Let K=(α1,,αn), and let 𝒪K denote the ring of integersMathworldPlanetmath of K. Then α1,,αn𝒪K and I𝒪K is an ideal of 𝒪K. Let h denote the class numberMathworldPlanetmathPlanetmath of K. Then (I𝒪K)h=β𝒪K for some β𝒪K. Let L=K(βh), and let 𝒪L denote the ring of integers of L. Then


Since unique factorizationMathworldPlanetmath of ideals holds in 𝒪L, I𝒪L=βh𝒪L. Since 𝒪K𝒪L and α1,,αnI𝒪KI𝒪L=βh𝒪L, there exist γ1,,γn𝒪L with αj=γjβh for all positive integers j with jn. Thus, I=α1,,αn=γ1βh,,γnβhβh. Since Iβh and I𝒪L=βh𝒪L, I=βh. Hence, I is principal. It follows that 𝔸 is a Bezout domain.

On the other hand, 𝔸 is not a principal ideal domainMathworldPlanetmath (PID). For example, the ideal all of the nth roots (http://planetmath.org/NthRoot) of 2, J=2,2,23,, is an ideal of 𝔸 that is not principal.

Title example of a Bezout domain that is not a PID
Canonical name ExampleOfABezoutDomainThatIsNotAPID
Date of creation 2013-03-22 16:57:04
Last modified on 2013-03-22 16:57:04
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Example
Classification msc 11R29
Classification msc 11R04
Classification msc 13G05