The main result we wish to prove is the following.
A ring A is artinian if and only if it is noetherian and , where denotes the Krull dimension.
Recall that means all prime ideals of A are maximal.
Since the proof is long we will break it up into steps.
Artinian ⟹ Dim = 0
Step 1: An artinian integral domain A is a field.
Let . From the sequence we have for some n > 0. Thus for some . Since A is a domain and we have so x is a unit.
Step 2: An artinian ring A has finitely many maximal ideals.
Let be the collection of ideals of A of the form where are maximal ideals. Since A is artinian has a minimal element . By minimality for any maximal ideal we have and so . We claim that for some which would complete our claim.
Since and are all maximal it suffices to show for some i. Indeed if not we can pick for . Then
Step 3: An artinian ring A has finitely many prime ideals, all maximal.
For any prime ideal , is an artinian integral domain, hence a field by step 1. Thus is maximal so all prime ideals of A are maximal. By step 2, there are only finitely many of them.
Thus, we have shown that an artinian ring has Krull dimension 0. Since the connected components of Spec A are all singleton sets, we have proven:
An artinian ring A is a finite product , where each is a local artinian ring with a unique prime ideal.
Our next target is noetherianness.
Artinian ⟹ Noetherian
By the intermediate result, we may assume A is a local artinian ring, with a unique prime ideal which is the nilradical of A. [ Recall (proposition 5 here) that in any ring, the nilradical is the intersection of all its prime ideals. ]
Step 4: is nilpotent.
We claim for some N > 0. Note that this is not a trivial result: since is the nilradical, for each we can find N such that but we have to find an N which works for all x.
For that take . Since A is artinian for some N>0. Write for this ideal so . It remains to show .
If not, among all ideals such that , pick a minimal . Pick any such that . Since minimality of forces . Also
by minimality of again. Thus for some . Since y is nilpotent we have for some N; hence , a contradiction.
Step 5: An artinian ring A is noetherian.
Continuing step 4, we now have
Each is an artinian A-module. Furthermore is a vector space over ; by exercise B.2 here this is artinian and hence finite-dimensional. Thus is a noetherian module over and hence over A (again by exercise B.2 here). This shows that A is noetherian.
Noetherian + Dim 0 ⟹ Artinian
Step 6: A noetherian ring A of Krull dimension 0 is artinian.
Since the connected components of Spec A are all singleton sets, A is a direct product of noetherian rings , where each has a unique prime ideal. It suffices to prove each is artinian so we assume A is noetherian and has a unique prime ideal .
Since A is noetherian, has a finite generating set . There is an M > 0 such that for each i. It follows that . [ Indeed this ideal is generated by all over all , . Since for some i, we have . ]
Let N = nM so that . As above, form the sequence
Each is now a noetherian vector space over so it is finite-dimensional. Thus it is also artinian over and hence over A. This shows that A is artinian. ♦
This completes our proof. Together with the previous article, we have:
Every artinian ring has a composition series and thus a well-defined length and set of composition factors.
Although all artinian rings are noetherian, there are artinian modules which are not noetherian, as we saw in the previous article.
For non-commutative rings, it is also true that a (left) artinian ring is (left) noetherian, but its proof is much more involved.
Now suppose A is a reduced artinian ring. We factor
as above, where each is an artinian ring with a unique prime ideal . Since each is also reduced, its nilradical is zero so is a field. Hence we have shown:
The ring A is reduced and artinian if and only if it is isomorphic to a finite product of fields.
We also have the following special case.
Let A be an algebra over a field k such that as a vector space. Then A is noetherian, and
where each has a unique prime ideal . Furthermore,
- is the number of prime ideals of A;
- if A is reduced then each is a finite field extension of k;
- if A is reduced and k is algebraically closed, then each so .
Note that in the context of algebraic geometry, if for an affine k-scheme V (k algebraically closed), then each corresponds to a point .