This is a problem I asked in SE, but it seems the question is more suitable for MO. 


Consider a ring $R$ (not necessary with identity or commutative) such that for any proper two-sided ideal $I$, $R\cong\frac RI$ (as rings), e.g., $\Bbb{Z}\_{2^\infty}$ with zero product. Is it true that the set of two-sided ideals of $R$ is a chain ?!



On rings $R$ for which $R \cong \frac RI$ for any proper two-sided ideal $I$
If there are ideals $I\_1,I\_2$ not forming a chain, we can assume that $I\_1,I\_2\ne0=I\_1\cap I\_2$ just passing to $R\simeq R/(I\_1\cap I\_2)$. Pick any $0\ne r\in R$. By the Zorn lemma, there is a maximal ideal $I$ in the set of the ideals with the property $r\notin I$. Passing to $R\simeq R/I$, we can assume that every nonnull ideal of $R$ contains $r$. So, $r\in I\_1\cap I\_2=0$. A contradiction. Therefore, the ideals of $R$ form a chain.


