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

There is an exact sequence of $R$-modulesIf $A$ is a flat module over $R$, tensoring this sequence with $A$ preserves its left exactness. The image of each tensor product inside $A$ is the corresponding extension of an ideal, soBoth displayed inclusions therefore hold. This is the flat extension preserves finite ideal intersections property.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.