There is an exact sequence of $R$-modules
$$
0\longrightarrow I\cap J
\xrightarrow{x\mapsto(x,x)}
I\oplus J
\xrightarrow{(x,y)\mapsto x-y}
I+J\longrightarrow0.
$$
If $A$ 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>, so
$$
\boxed{I^e\cap J^e=(I\cap J)^e}.
$$
Both displayed inclusions therefore hold. This is the <flat extension preserves finite ideal intersections> property.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$I^e$}
{wiki=Extension_of_ideals}

For a ring homomorphism $f:R\to A$ and an ideal $I\subseteq R$, the extension $I^e=IA$ is the ideal of $A$ generated by $f(I)$.


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 Incoming links: Extension of an ideal