half-factorial ring

An integral domainMathworldPlanetmath D is called a half-factorial ring (HFD) if it satisfies the following conditions:

  • Every nonzero element of D that is not a unit can be factored into a product of a finite number of irreducibles.

  • If  p1p2pm  and  q1q2qn  are two factorizations of the same element a into irreducibles, then  m=n.

If, in , the irreducibles pi and qj are always pairwise associatesMathworldPlanetmath, then D is a factorial ring (UFD).

For example, many orders (http://planetmath.org/OrderInAnAlgebra) in the maximal orderMathworldPlanetmath of an algebraic number fieldMathworldPlanetmath are half-factorial rings, e.g. [32] is a HFD but not a UFD (see http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdfthis paper).

Title half-factorial ring
Canonical name HalffactorialRing
Date of creation 2013-03-22 18:31:14
Last modified on 2013-03-22 18:31:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 13G05
Synonym half-factorial domain
Defines HFD