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?