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principal_ideal_domains [2010/07/29 15:46] (current)
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 +=====Principal Ideal Domain=====
 +Abbreviation: **PIDom**
 +====Definition====
 +A \emph{principal ideal domain} is an [[integral domains]] $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ in which
 +
 +
 +every ideal is principal:  $\forall I \in Idl(R)\ \exists a \in R\ (I=aR)$
 +
 +Ideals are defined for [[commutative rings]]
 +
 +==Morphisms==
 +====Examples====
 +Example 1: ${a+b\theta  | a,b\in Z, \theta=\langle 1+ \langle-19\rangle^{1/2}\rangle/2}$ is a Principal Ideal Domain that is not an [[Euclidean domains]]
 +
 +See Oscar Campoli's "A Principal Ideal Domain That Is Not a Euclidean Domain" in <i>The American Mathematical Monthly</i> 95 (1988): 868-871
 +
 +
 +====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====
 +[[Euclidean domains]]
 +
 +====Superclasses====
 +[[Unique factorization domains]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]