Commutative Algebra/Artinian rings

Definition, first property

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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

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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  .