If every prime ideal is maximal, what can we say about the ring?

For a commutative ring, having all primes maximal has a simple characterization: $R/J(R)$ is von Neumann regular and $J(R)$ is a nil ideal, where $J(R)$ is the Jacobson radical.

This recovers everything previously mentioned:

• When $R$ is Artinian, $R/J(R)$ is semisimple (hence VNR) and $J(R)$ is nilpotent (hence nil.)

• When $R$ is VNR, then $J(R)=\{0\}$ (hence nil) and $R/J(R)=R$ is obviously VNR.

• When $R$ is Noetherian, a nil $J(R)$ becomes a nilpotent $J(R)$, and a Noetherian $R/J(R)$ is necessarily semisimple, so Hopkins-Levitzki says that $R$ is Artinian.

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I'm not aware of a definitive answer for noncommutative rings. Things are different there because simple rings are a lot more diverse than fields, prime ideals are less nice, localization is not nice, and nilpotent elements don't necessarily live in $J(R)$ anymore.

Here's a fitting generalization I found. The above theorem for commutative rings is "$R/J(R)$ is VNR and $J(R)$ is nil iff $R/P$ is a field for every prime ideal $P$," and the following is also true: "$R/P$ is a division ring for all prime ideals $P$ iff $R/J(R)$ is strongly regular and $J(R)$ is nil."

If we assume $R$ is commutative and Noetherian, then this property is equivalent to $R$ being an Artinian ring (i.e., satisfying the descending chain condition). Such rings are finite products of Artin local rings.

Reduced Artin local rings are fields. Some non-reduced examples include $k[x]/(x^n)$, $k$ a field, and more generally $k[x_1,\ldots,x_n]/I$, where Rad$(I)=(x_1,\ldots,x_n)$. There are also examples that don't contain a field, like $\mathbb{Z}/(p^n)$, $p$ a prime.

I assume $R$ is commutative. Such a ring is said to have Krull dimension $0$ or to be zero-dimensional.

• Every field is zero-dimensional. More generally, every Artinian local ring is zero-dimensional.
• A (edit: finite) product of zero-dimensional rings is zero-dimensional. In particular, every (edit: finite) product of Artinian local rings is zero-dimensional.
• Every Boolean ring is zero-dimensional. This gives a supply of examples that are in general neither Noetherian nor products of Artinian local rings.
• According to Wikipedia, zero-dimensional and reduced is equivalent to von Neumann regular.

I don't think there is a nice classification of arbitrary rings of Krull dimension $0$ (and I have no idea what happens in the noncommutative case).