# Differences

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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]] | ||

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+ | 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 | ||

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+ | |||

+ | ====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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