What kind of rings have exactly three ideals?

Question

What kind of rings(commutative, w/ unity) have exactly three ideals? I know that those with exactly two ideals are "the fields", but what about three? Is there a fancy name for them?

Answer 1

Let $R$ be a commutative with exactly three ideals. These must be $0 \subset \mathfrak{m} \subset R$ for some ideal $\mathfrak{m}$. It follows $\mathfrak{m}$ is a maximal ideal, in fact the unique one, so that $R$ is local.

Choose $x \in \mathfrak{m} \setminus \{0\}$, then $0 \neq \langle x \rangle \neq R$, hence $\mathfrak{m} = \langle x \rangle$. We conclude that $R$ is a special principal ideal ring. Now look at $\mathfrak{m}^2$. If $\mathfrak{m}^2=\mathfrak{m}$, then the Nakayama Lemma implies $\mathfrak{m}=0$, a contradiction. Otherwise we have $\mathfrak{m}^2=0$.

Conversely, let $R$ be a special principal ideal ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^2=0$ and $\mathfrak{m} \neq 0$. Then $R$ has exactly three ideals.