Question

A few questions relevant formally, but quite different in nature:

From now on, let R denote a ring.

1. If R is a UFD , is R[x] also a UFD?

2. If R is Noetherian, is R[x] also Noetherian?

3. If R is a PID, is R[x] also a PID?

4. If R is an Artin ring, is R[x] also an Artin ring?

For 1, we all know it's Gauss's lemma.

For 2, we all know it's Hilbert's basis theorem.

For 3, we all know that in Z[x], the ideal (2,x) is not a principal ideal, so the answer is negative.

But what about 4?

Answer 1

Good argument. But let's give a down-to-earth counterexample:

Let $R$ be a field. Then consider $$(x)\supset (x^2)\supset(x^3)\supset\ldots.$$ This is a descending chain of ideals that doesn't become stationary so $R[X]$ is not Artin.

Answer 2

The answer to 4 is "no." If $R$ is an Artin ring, then it is Noetherian of Krull dimension zero. It follows from dimension theory that $R[X]$ is Noetherian of dimension one, i.e., not every prime ideal in $R[X]$ is maximal, so $R[X]$ can't be Artin.