# Differences

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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> | ||

+ | )] |

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