half-factorial ring

An integral domain $D$ 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 $p_{1}p_{2}\\cdots p_{m}$ and $q_{1}q_{2}\\cdots q_{n}$ are two factorizations of the same element $a$ into irreducibles, then $m=n$ .

If, in , the irreducibles $p_{i}$ and $q_{j}$ are always pairwise associates, then $D$ is a factorial ring (UFD).

For example, many orders (http://planetmath.org/OrderInAnAlgebra) in the maximal order of an algebraic number field are half-factorial rings, e.g. $\\mathbb{Z}[3\\sqrt{2}]$ is a HFD but not a UFD (see http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdfthis paper).

单词	HalffactorialRing
释义	half-factorial ring An integral domain $D$ 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 $p_{1}p_{2}\\cdots p_{m}$ and $q_{1}q_{2}\\cdots q_{n}$ are two factorizations of the same element $a$ into irreducibles, then $m=n$ . If, in , the irreducibles $p_{i}$ and $q_{j}$ are always pairwise associates, then $D$ is a factorial ring (UFD). For example, many orders (http://planetmath.org/OrderInAnAlgebra) in the maximal order of an algebraic number field are half-factorial rings, e.g. $\\mathbb{Z}[3\\sqrt{2}]$ is a HFD but not a UFD (see http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdfthis paper).
随便看	development Deviance deviance Diagonal diagonal DiagonalEmbedding diagonal embedding DiagonalIntersection diagonal intersection Diagonalizable diagonalizable DiagonalizableOperator diagonalizable operator Diagonalization diagonalization DiagonalizationOfQuadraticForm diagonalization of quadratic form DiagonallyDominantMatrix diagonally dominant matrix DiagonalMatrix diagonal matrix DiagonalQuadraticForm diagonal quadratic form Diameter diameter