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

From now on, let R denote a ring.

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

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

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?