# Noetherian induction

A reasoning principle applicable to a partially ordered set in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some 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.

#### References

[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |

#### Comments

The term well-founded induction is also in use.

**How to Cite This Entry:**

Noetherian induction.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Noetherian_induction&oldid=32633