Let $R$ be a commutative ring with $1$ and $I$ an ideal. Also let $B$ be a principal ideal, and $A=\{a\in R\;|\; aB\subseteq I\}$. I want to show that if $A$ is also principal then $I$ is principal.
Have $A=(a),\;B=(b)$. I first thought $I=R$ or $I=(ab)$: certainly $(ab)\subseteq I$ since $ r\cdot(ab)=a\cdot(rb)\in I$, but whether $I\subseteq (ab)$ I am not sure. Either way I'm not sure how to proceed, usually I show an ideal is principal by picking an special element (irreducible, minimal...), but there doesn't seem to be much choice here.
Any help appreciated.
You can't prove what you want for it is not true.
Let $I$ be a non-principal ideal, and $b\in I$. Then for $B=(b)$ we have $A=I:(b)=(1)$, a principal ideal.