Does there exist a ring which is not a principal ideal ring and which has exactly six different ideals?
(For me a ring is commutative with a unit element.)
I can show that any ring having at most five ideals is a principal ideal ring.
EDIT: The proof goes as follows, it is not so hard. Suppose that $R$ is a ring which is not principal and which has at most five ideals. Then there must exist a proper ideal of the form $(\alpha,\beta)$ in $R$ which is not principal. Then $(0), (\alpha),(\beta),(\alpha, \beta), R$ must be the five different ideals of $R$. But what about $(\alpha + \beta)$? It cannot be $(0)$ or $(\alpha,\beta)$ or $R$. Say that it is $(\alpha)$. Then we get that $\alpha \mid \beta$, hence $(\alpha,\beta) = (\alpha)$, contradiction! Similarly, it cannot be $(\beta)$. Hence we are done.