example of ring which is not a UFD

Example 1.

We define a ring R=[-5]={n+m-5:n,m} with addition and multiplication inherited from (notice that R is the ring of integersMathworldPlanetmath of the quadratic number field (-5)). Notice that the only units (http://planetmath.org/UnitsOfQuadraticFields) of R are R×={±1}. Then:

6=23=(1+-5)(1--5). (1)

Moreover, 2, 3, 1+-5 and 1--5 are irreducible elementsMathworldPlanetmath of R and they are not associatesMathworldPlanetmath (to see this, one can compare the norm of every element). Therefore, R is not a UFD.

However, the ideals of R factor (http://planetmath.org/DivisibilityInRings) uniquely into prime idealsMathworldPlanetmathPlanetmathPlanetmath. For example:


where 𝔓=(2,1+-5), 𝔔=(3,1+-5), and 𝔔¯=(3,1--5) are all prime ideals (see prime ideal decomposition of quadratic extensions of (http://planetmath.org/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ)). Notice that:


Thus, Eq. (1) above is the outcome of different rearrangements of the product of prime ideals:


Notice also that if 𝔓 was a principal idealMathworldPlanetmathPlanetmath then there would be an element αR with (α)=𝔓 and (α)2=(2). Thus such a number α would have norm 2, but the norm of n+m-5 is n2+5m2 so it is clear that there are no algebraic integersMathworldPlanetmath of norm 2. Therefore 𝔓 is not principal. Thus R is not a PID.

Title example of ring which is not a UFD
Canonical name ExampleOfRingWhichIsNotAUFD
Date of creation 2013-03-22 15:08:19
Last modified on 2013-03-22 15:08:19
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Example
Classification msc 13G05
Synonym example of a ring of integers which is not a UFD
Related topic DeterminingTheContinuationsOfExponent
Defines example of a number ring which is not a UFD