# Are all Artinian Rings Jacobson?

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.

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

• I meant to write ideal agree, i.e. for all ideals I, r(I) ,the radical of I equals the Jacobson radical of I Commented May 3, 2022 at 14:58
• Oh ha, i see, then yes Commented May 3, 2022 at 14:59
• I'm not aware of any standard notation for the Jacobson radical of an ideal Commented May 3, 2022 at 14:59
• commalg.subwiki.org/wiki/Jacobson_ring Commented May 3, 2022 at 15:04

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.
• Actually it turns out you only need the DCC on chains of the form $xR\supseteq x^2R\supseteq x^3R\supseteq \ldots$ to get it to be Jacobson Commented May 3, 2022 at 17:03