Definition, first property
A ring is called artinian if and only if each descending chain
of ideals of eventually terminates.
Equivalently, is artinian if and only if it is artinian as an -module over itself.
Let be an artinian integral domain. Then is a field.
Let . Consider in the descending chain
Since is artinian, this chain eventually stabilizes; in particular, there exists such that
Then write , that is, , that is (as we are in an integral domain) and has an inverse.
Let be an artinian ring. Then each prime ideal of is maximal.
If is a prime ideal, then is an artinian (theorem 12.9) integral domain, hence a field, hence is maximal.
Let be a ring. We have:
- is artinian is noetherian and every prime ideal of is maximal.
First assume that the zero ideal of can be written as a product of maximal ideals; i.e.
for certain maximal ideals . In this case, if either chain condition is satisfied, one may consider the normal series of considered as an -module over itself given by
Consider the quotient modules . This is a vector space over the field ; for, it is an -module, and annihilates it.
Hence, in the presence of either chain condition, we have a finite vector space, and thus has a composition series (use theorem 12.9 and proceed from left to right to get a composition series). We shall now go on to prove that is a product of maximal ideals in cases
- is noetherian and every prime ideal is maximal
- is artinian.
1.: If is noetherian, every ideal (in particular ) contains a product of prime ideals, hence equals a product of prime ideals. All these are then maximal by assumption.
2.: If is artinian, we use the descending chain condition to show that if (for a contradiction) is not product of prime ideals, the set of ideals of that are product of prime ideals is inductive with respect to the reverse order of inclusion, and hence contains a minimal (w.r.t. inclusion) element . We lead this to a contradiction.
We form . Since as , . Again using that is artinian, we pick minimal subject to the condition . We set and claim that is prime. Let indeed and . We have
- , hence, by minimality of ,
and similarly for . Therefore
whence . We will soon see that . Indeed, we have , hence and therefore
This shows , and contradicts the minimality of .