Suppose $R$ is a commutative ring, not necessarily Dedekind. It has an ideal $I$, not necessarily principal, and $I\neq I^2\neq I^3$. Is it always possible to find $a,b\in I$ so $ab\notin I^3$?
I know I $must$ choose $a,b\notin I^2$; however, for a non-Dedeking ring that's not sufficient. What tools can I use, to make sure $ab\notin I^3$?
If $xy\in I^3$ for every $x,y\in I$, then all the generators of $I^2$ are in $I^3$, therefore $I^2\subseteq I^3\subseteq I^2$.
By contrapositive then, if one assumes $I^2\neq I^3$, it must be that there exists $x,y\in I$ such that $xy\notin I^3$.