Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.

Let $R$ be a finite commutative ring with identity. Under what conditions the number of ideals of $R$ is equal to the number of elements of $R$?

The only class of rings with this property that I know is the class of finite boolean rings. I do not know if the converse is true. So any suggestion would be helpful.