Part 11 of 13. Every ascending chain of ideals eventually stops. Noether made this an axiom in 1921.
The ideal factorization of Part 10 rests on a condition Dedekind stated in 1894 for rings of algebraic integers and Noether made an axiom for all commutative rings:
A ring with this ascending chain condition is Noetherian. The rest of Act III runs on it.
Every ideal of is some , and exactly when . So the ideals above are its divisors, a finite set for a chain to climb.
Figure 1. Every ideal containing ; each edge divides out one prime. Click an ideal above the chain to climb to it, or the start to reset (keyboard: arrows and Enter). Greyed ideals are out of reach.
Each strict step divides out at least one prime, so a chain from has at most strict steps, the number of prime factors with multiplicity. The readout finds the longest path in the diagram, and it always equals that.
over a field is also a principal ideal domain. Adding to gives , and a strict step lowers the degree, so a chain from has at most strict steps.
Figure 2. The current generator (violet) against the start (grey), with Euclid run in exact rational arithmetic. Dots under the axis are roots, stacked by multiplicity; a strict step removes at least one, and there are four.
In two variables needs two generators, and the gcd argument is gone. Draw as the cell : a monomial ideal is a staircase closed up and to the right. Dickson's lemma says a staircase has finitely many inner corners, so a growing sequence of staircases stops.
Figure 3. Click a cell to add that monomial (keyboard: arrows and Enter). Filled cells are in the ideal, outlined ones are its minimal generators, and the readout counts the monomials still outside.
Once the staircase touches both axes, the monomials outside are finite and each strict step fills one, so their count is a budget. Before that there is none: is as long as you like. Every chain stops, but nothing says when.
Three conditions on a ring are equivalent: (a) every ascending chain of ideals stabilizes; (b) every ideal is finitely generated; (c) every nonempty set of ideals has a maximal element.
For (b) ⇒ (a): the union of a chain is an ideal with finitely many generators, each appears at some step, and after the last one the chain cannot move. For (a) ⇒ (b): if has no finite generating set, pick , , , and so on forever. (a) and (c) are one statement read two ways: a set with no maximal element lets you climb forever, and an endless chain is such a set.
Figure 4. Random monomials of degree at least 3, added one at a time. Green: each factor is or . Red: each factor is with probability . Curves count strict steps (square-root scale); ticks mark when each generator of the green union arrived.
Every ring so far is Noetherian, as is every ring of integers of a number field. The red stream's ring is not: never stops, since setting kills but not .
The ring of all algebraic integers fails too: , because each quotient is a root of the monic but its inverse, a root of , is not an algebraic integer.
Figure 5. Ideals containing , from at to the whole ring at . Row is the ring of integers of , where and these ideals are the ; the bottom row is all algebraic integers, and the dashed path is the chain above. Click a row or use the arrow keys.