It is well-known that, for a commutative ring $A$ which is not a PID, there may exist a triple of ideals, $0\neq I, J_1, J_2$ such that $J_1\neq J_2$ but $IJ_1=IJ_2$. I would like to present an explicit example of this fenomenon to my students at a basic undergraduate course in commutative algebra. Do you have any suggestion?
Essentially the unique examples they will understand will be about polinomial rings and matrix rings.
