Question

Prove that every field is artinian.Deduce that Q is artinian.

Answer 1

Let us prove that every field is Artinian. Recall that an Artinian ring is a ring that satisfies the descending chain condition on ideals, i.e. for any decreasing sequence of right ideals $I_1\supset I_2\supset\ldots$ there exists a natural number m such that $I_m=I_{m+1}=\ldots$ .

Taking into account that each field $F$ contains exactly two ideals: $F$ and $O=\{0\}$, we conclude that for each descending chain of ideals there are two possibilites:

1) Each ideal in a chain is equal to $F$.

2) There exists an ideal $I_m$ that is equal to $O$. Since we consider only descending chains, $I_k=O$ for every integer $k>m$.

We conclude that in each case a field satisfies the descending chain condition on ideals, and hence each field is Artinian.

Since $(\mathbb Q,+,\cdot)$ is a field, it is Artinian.