Commutative Algebra/Artinian rings

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Definition, first property[edit]

Definition 19.1:

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.

Proposition 19.2:

Let be an artinian integral domain. Then is a field.

Proof:

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.

Corollary 19.3:

Let be an artinian ring. Then each prime ideal of is maximal.

Proof:

If is a prime ideal, then is an artinian (theorem 12.9) integral domain, hence a field, hence is maximal.

Characterisation[edit]

Theorem 19.4:

Let be a ring. We have:

is artinian is noetherian and every prime ideal of is maximal.

Proof:

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

  1. is noetherian and every prime ideal is maximal
  2. 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 .