We will assume that is a commutative ring with unity and the polynomial ring over We will denote by the Jacobson radical of

**Problem 1.** The ring is never Artinian.

**Solution**. Let the ideal of generated by the coset

*Claim* . for all integers

*Proof of the claim* . Suppose, to the contrary, that for some Then

and so for some Hence and therefore must be a unit in But the coefficient of in is , which is obviously not nilpotent, and so cannot be a unit (see here). Contradiction!

So, by the claim, we have a strictly descending chain of ideals of proving that is not Artinian.

**Problem 2**. Prove that has infinitely many maximal ideals.

**Solution**. Suppose, to the contrary, that the set of maximal ideals of is finite. Let be the maximal ideals of Then, by the Chinese remainder theorem, So, since each is a field and fields have only two ideals, must have finitely many ( in fact) ideals. But then would obviously be Artinian, contradicting Problem 1.

Suppose that is Noetherian. Then by the Hilbert’s basis theorem, is Noetherian too. Thus is Noetherian. So is always a non-Artinian Noetherian ring.