Does anyone know where this problem is taken from:
Prove that if $I$ is a radical ideal (in a commutative ring) and $ab∈I$, then $$I=\operatorname{rad}(I+(a))∩\operatorname{rad}(I+(b)).$$
I found it in Every radical ideal in a Noetherian ring is a finite intersection of primes.
This would be a straightforward way to prove that every radical ideal in a Noetherian ring is a finite intersection of prime ideals, but I have not been able to find any reference that uses any approach similar to this.