What other classes of commutative rings can be defined by requiring that $\{0\}$ is the only proper ideal satisfying some condition?

Question

A field is just a commutative ring $R$ such that $\{0_R\}$ is the only proper ideal. Interestingly, there's a similar characterization of integral domains. Given a subset $A$ of $R$, let $A^\perp$ denote the annihilator of $A$, i.e. the set of all $x \in A$ such that $xA \subseteq \{0_R\}$. Then $A^\perp$ is always an ideal. Call a subset $I$ of $R$ an annihilator iff $I=A^\perp$ for some $A \subseteq R$; equivalently, iff $I^{\perp\perp} = I$. Then an integral domain is just a commutative ring $R$ such that $\{0_R\}$ is the only proper annihilator.

Question. What other interesting classes of commutative rings can be defined by requiring $\{0\}$ is the only proper ideal satisfying some condition?

Answer 1

A ring $R$ with exactly two non-maximal ideals, $\{0\}$ and $R$, is either a local ring that's not a field or a product of two fields.

Answer 2

A commutative ring is connected (no nontrivial idempotents) iff $\{0\}$ and $R$ are the only finitely generated idempotent ideals.

Near misses:

A commutative ring is semisimple iff $R$ is the only essential ideal.

A commutative ring has trivial Jacobson radical if $\{0\}$ is the only superfluous ideal.

A commutative ring is reduced iff $\{0\}$ is the only nilpotent ideal.