A commutative ring with at most $5$ distinct ideals is a PIR

Question

Let $R$ be a commutative ring with unity having at most $5$ distinct ideals (including $\{0\}$ and $R$ itself); then is it true that $R$ is a principal ideal ring i.e. is every ideal of $R$ principal?

Answer 1

Suppose $R$ has exactly $5$ ideals and one is nonprincipal, call it $I$.

Of course the nonprincipal one still has to be finitely generated (the ring is Artinian), so select a minimal generating set $\{a, b, \ldots\}$.

How many ideals are we faced with already? There is certainly $\{0\}$, $I$, and $R$, as well as $(a)$, $(b)$ which are all certainly mutually distinct because you picked the generating set to be minimal.

But now consider this: which one of these ideals is the principal ideal $(a+b)$?