# Jacobson radical and prime ideal

I am looking for an equivalent condition for a commutative ring $R$ with 1 to have the following property:

The only prime ideals contain $J (R)$, jacobson radical of $R$, are maximal ideals.

• Are you asking what type of rings have the property that the only prime ideals containing $J(R)$ are maximal? – rschwieb Mar 1 '17 at 19:16

You are asking simply that $R/J(R)$ has Krull dimension zero.

This occurs exactly when $R/J(R)$ is von Neumann regular.

It follows from a more general theorem that says this:

In a commutative ring $R$, the following are equivalent:

1. $R$ is zero dimensional
2. $J(R)$ is a nil ideal and $R/J(R)$ is von Neumann regular.

Of course in your case, $J(R/J(R))$ is nil (since it is zero.)

• There are a ton of equivalent conditions to be von Neuamann regular. I'd prefer not to list them all. But maybe if you have some additional information about what characterization would make the most sense you can let me know. – rschwieb Mar 1 '17 at 19:27