unique_factorization_domains

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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>
)]

Trace: • commutative_ordered_rings • semilattices_with_identity • neardistributive_lattices • 2-element_boolean_algebra • finite_commutative_monoids • vector_spaces_over_a_field • integral_relation_algebras • compact_topological_spaces • partially_ordered_monoids • finite_ordered_semigroups