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.
Sasha Anan'in
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