 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-2/21f/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

The equality is false. In

A = k [x, y]

, take

I = (y)

and

J = (y - x^{2})

. Both are prime ideals, since their quotient rings are isomorphic to

k [x]

, so both are radical. But

I + J = (y, x^{2}), I + J = (x, y) .

The element

x

belongs to the latter ideal and not the former. This also shows why taking the radical of an ideal can enlarge a sum of already radical ideals.

 New to topics? Read the docs here!