Are all Artinian Rings Jacobson?

Question

An Artinian Ring is one which satisfies the descending chain condition for ideals.

A Jacobson Ring is one where the radical and Jacobson radical of an ideal agree, for all ideals.

(Assuming Ring = Unital & Commutative here)

We know the following about Artinian Rings:

• Quotients of an Artinian are Artinian.
• The Nilradical and Jacobson Radical of an Artinian Ring agree.

Putting these 2 together I believe we can deduce all Artinian Rings are Jacobson.

However, If this result is indeed true, I'm surprised it wasn't mentioned in lectures or find reference to it in any books.

If it doesn't hold, is there a counterexample?

Thanks

Henry

Answer 1

By this notion I think you mean the well-known definition for commutative rings: every prime ideal is an intersection of maximal ideals.

It is equivalent to $J(R/I)=N(R/I)$ for all ideals $I\lhd R$.

Equivalently, that means the intersection of primes over $I$ (the "radical of $I$" and the intersection of maximal ideals over $I$ (the "Jacobson radical of $I$") coincide.

In an Artinian ring, a prime ideal is maximal, so each prime is trivially an intersection of a single maximal ideal. That means that commutative Artinian rings are indeed Jacobson.