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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.

user26857
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Algebrica
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  • I suppose that you are looking for a proof, not for a reference of the highlighted result. Anyway, the proof is straightforward. – user26857 Apr 28 '18 at 08:42
  • No, I am looking for a reference. The proof is really straightforward, but I have not been able to find this little lemma (the highlighted result) anywhere. The other approaches use primary decomposition, which is too complicated for my purposes. I cannot claim I have come up with the lemma on my own nor can I refer to stackexchange... – Algebrica Apr 29 '18 at 09:34
  • Then please use the appropriate tag reference-request. – user26857 Apr 29 '18 at 09:53
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    @Algebrica: Since the proof is straightforward, why not just write down the proof (or even just write that this is straightforward!)? You don't need reference for every single small fact you use in mathematical work. When you're using nontrivial theorems, you should give credit where it is due. With simple stuff like this, you can just write it as a proposition or a fact (maybe noting that it is folklore). – tomasz Apr 29 '18 at 10:12
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    And, well, you can refer to stackexchange. See: https://math.meta.stackexchange.com/questions/8212/citing-stackexchange-postings . – tomasz Apr 29 '18 at 10:13
  • Well, personally I like better referring to stackexchange than claiming that I came up with the approach myself. Thanks! – Algebrica Apr 30 '18 at 11:15

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