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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>
+)]

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