principal_ideal_domains - MathStructures

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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>
)]

Trace: • bounded_distributive_lattices • terms • cancellative_commutative_monoids • lattices • sequential_algebras • equivalence_relations • representable_cylindric_algebras • lukasiewicz_algebras_of_order_n • jordan_algebras • regular_semigroups