Definition:Well-Ordered Integral Domain
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Definition
A well-ordered integral domain is an ordered integral domain $\left({D, +, \times \le}\right)$ in which the ordering $\le$ induced by the positivity property is a well-ordering.
That is, every subset $S$ of the positive elements of $D$ has a minimal element:
- $\forall S \subseteq D_+^*: \forall a \in S: \exists x \in S: x \le a$
where $D_+^*$ denotes all the elements $d \in D$ such that $P \left({d}\right)$.
Sources
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 2.8$
