If for any two principal ideals one contains another, then for any two ideals one ideal contains another

Question

Let $R$ be a commutative ring with identity. Assume that for any two principal ideals $Ra$ and $Rb$ we have either $Ra\subseteq Rb$ or $Rb\subseteq Ra$. Show that for any two ideals $I$ and $J$ in $R$, we have either $I\subseteq J$ or $J\subseteq I$.

Initially i thought that if i could show that any ideal in the ring is principal then i am done. But could not show what i thought of. Is my assumption to solve the problem correct? How can i proceed? Any hints would be highly appreciated. Thank you.

Answer 1

I'm not sure trying to show every ideal is principal will work (though I can't verify a counterexample off the top of my head!) however I'll start you off a different way:

First assume $I \not\subseteq J$, then we can take some $x\in I\smallsetminus J$ and now consider the principal ideal $Rx$, what can we say about this ideal?

Answer 2

Suppose $I\not\subseteq J$ and $J\not\subseteq I$ you have $x\in I\setminus J$ and $y\in J\setminus I$

Now consider Principal ideals $Rx$ and $Ry$