# Commutative Algebra/Artinian rings

## 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

- 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 .