Let $I$ be a minimal right ideal of a ring $R$ with $1$. If $r\in R$, could we say that $rI$ is zero or a minimal right ideal?
I assumed a right ideal $J$ in $rI$ and intersecting it with $I$ got a right ideal in $I$ which is equal to zero or equal to $I$ (due to minimality). If the intersection equals $I$, we get $I⊆rI$. But, I could not proceed any more, and would thank for any contribution!
Intuitively, $rI$ might be very different from $I$ in terms of its members, so intersecting its subset with $I$ might not be useful.
If $J$ is a right ideal in $rI$, then, being a subset of $rI$, it has to arise as $rX$ for some $X\subset I$. Can you prove that $X$ is a right ideal? You might need to tighten the specification of $X$ a little to make it work.