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unique_factorization_domains [2010/07/29 15:46] (current)
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 +=====Unique Factorization Domains=====
 +Abbreviation: **UFDom**
 +====Definition====
 +A \emph{unique factorization domain} is an [[integral domains]] $D$ such that
 +
 +
 +every element is a product of irreducibles:  $\forall a\in D \exists p_1,...,p_r\in D, n_1,...,n_r\in \mathbb{N}$ such that $a=p_1^{n_1}\cdotp_2^{n_2}...p_r^{n_r}$ and
 +$p_i$ is irreducible for $i=1,\ldots,r$
 +
 +
 +the product is unique up to associates:  $\forall \mbox{ irreducibles } p_i,q_j$ if
 +$a=p_1^{n_1}\cdot p_2^{n_2}...p_r^{n_r}=q_1^{m_1}\cdot q_2^{m_2}...q_s^{m_s}$
 +then $r=s$ and each $p_i$ is an associate of some $q_j$
 +
 +==Morphisms==
 +
 +====Examples====
 +Example 1: $\mathbb{Z}[x]$, the ring of polynomials with integer coefficients.
 +
 +
 +====Basic results====
 +
 +====Properties====
 +^[[Classtype]]  |second-order |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  | |
 +^[[Residual size]]  | |
 +^[[Congruence distributive]]  | |
 +^[[Congruence modular]]  | |
 +^[[Congruence n-permutable]]  | |
 +^[[Congruence regular]]  | |
 +^[[Congruence uniform]]  | |
 +^[[Congruence extension property]]  | |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]  | |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &1\\
 +f(4)= &1\\
 +f(5)= &1\\
 +f(6)= &0\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Principal Ideal Domains]]
 +
 +====Superclasses====
 +[[Integral domains]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]