# Differences

This shows you the differences between two versions of the page.

unique_factorization_domains [2010/07/29 15:46] (current)
Line 1: Line 1:
+=====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>
+)]

##### Toolbox 