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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>
		+	)]