View source for Noetherian induction

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A reasoning principle applicable to a [[Partially ordered set|partially ordered set]] in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some [[Noetherian space|Noetherian space]]. Let $M$ be such a set and let $F$ be a subset of it having the property that for every $a\in F$ there is a strictly smaller element $b\in F$. Then $F$ is empty. For example, let $M$ be the set of all closed subsets of a Noetherian space and let $F$ be the set of those closed subsets that cannot be represented as a finite union of irreducible components. If $Y\in F$, then $Y$ is reducible, that is, $Y=Y_1\cup Y_2$, where $Y_1$ and $Y_2$ are closed, both are strictly contained in $Y$ and at least one of them belongs to $F$. Consequently, $F$ is empty.

Reversal of the order makes it possible to apply Noetherian induction to partially ordered sets in which every non-empty subset contains a maximal element; for example, to the lattice of ideals in a [[Noetherian ring|Noetherian ring]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>



====Comments====
The term well-founded induction is also in use.