commutative ring which have every maximal ideal generated by an idempotent

Question

Can you help me with one example of commutative ring which have every maximal ideal generated by an idempotent?

Answer 1

What about the following (non-trivial?) example: $R=\mathbb C\times\mathbb C$?

One can prove the following:

If $R$ is a commutative ring such that all its maximal ideals are generated by an idempotent, then $R$ is isomorphic to a finite direct product of fields.

For a proof see here.

Answer 2

Let $R$ be a field. then its only maximal ideal is $0.$

---

A ring is semisimple if and only if every right (or every left) ideal is generated by an idempotent. If you want to know semisimple rings, note that a ring is semisimple if and only if it is Artinian and its Jacobson radical is zero.