How do you prove that 2 nontrivial ideals are the only nontrivial ideals in a ring?

Question

What do you need to look for in 9rder to prove that 2 nontrivial ideals are the only nontrivial ideals in the ring? I can prove that if my ideals are I and J that I+J=R, where R is the ring, but I don't think this is enough in general. I think that the Chinese theorem has something to do with it.

Answer 1

Depends on what you mean by „trivial“. If you mean non-zero, then $R$ is a ring with exactly three ideals. Thus $R$ is a local ring and the ideals are $(0) \subset \mathfrak{m} \subset R$, where $\mathfrak{m}$ is the unique maximal ideal of $R$. To prove that a ring is local you can show that every element not being in the maximal ideal is a unit.

Since you were saying something about coprime ideals I believe you rather want to show that $R$ has exactly $4$ ideals. What is the ring you are considering?