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Revision difference : half-factorial ring |
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An integral domain $D$ is called a {\em half-factorial ring} (HFD) if it satisfies the following conditions:
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An integral domain $D$ is called a {\em half-factorial domain} (HFD) if it satisfies the following conditions:
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| \begin{itemize} |
\begin{itemize} |
| \item Every nonzero element of $D$ that is not a unit can be factored into a product of a finite number of irreducibles. |
\item Every nonzero element of $D$ that is not a unit can be factored into a product of a finite number of irreducibles. |
| \item If\, $p_1p_2\cdots p_m$\, and\, $q_1q_2\cdots q_n$\, are two factorizations of the same element $a$ into irreducibles, then\, $m = n$. |
\item If\, $p_1p_2\cdots p_m$\, and\, $q_1q_2\cdots q_n$\, are two factorizations of the same element $a$ into irreducibles, then\, $m = n$. |
| \end{itemize} |
\end{itemize} |
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| If, in \PMlinkescapetext{addition}, the irreducibles $p_i$ and $q_j$ are always pairwise associates, then $D$ is a factorial ring (UFD).\\ |
If, in \PMlinkescapetext{addition}, the irreducibles $p_i$ and $q_j$ are always pairwise associates, then $D$ is a factorial ring (UFD).\\ |
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For example, many \PMlinkname{orders}{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 \PMlinkexternal{this paper}{http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdf}).
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For example, many \PMlinkname{orders}{OrderInAnAlgebra} in the maximal order of an algebraic number field are half-factorial domains, e.g. $\mathbb{Z}[3\sqrt{2}]$ is a HFD but not a UFD (see \PMlinkexternal{this paper}{http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdf}).
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