Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 1 / v / b / Solution

The inclusionalways holds: each generator coming from $I\cap J$ lies in both extended ideals.

The reverse inclusion can fail. Letfor a field $k$. Under $A\cong k[x]$,whereas $I\cap J=(xy)$ in $R$, soThus statement (2) is true in general and statement (1) is false in general.