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Could someone show the reference of this rule: if $I = (ab) + J$, then $I = ((a) + J) \cap ((b) + J)$.

I have found it here.

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Zirko Tammado
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    Not quite true. In $\Bbb Z$ take $J=(0)$ and $a=6$, $b=10$. Then $((a)+J)\cap((b)+J)=(30)\ne(60)$. You need $a$ and $b$ to be coprime. – anon Jan 08 '14 at 20:57
  • I will use it for $I=(xy,xz,yz)$ in $\Bbb R[x,y,z]$ and I believe it will be true (correct me if I am wrong), but I need the reference (the book that containing this rule). – Zirko Tammado Jan 08 '14 at 21:04

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I suppose that $I,J$ are monomial ideals, and $a,b$ monomials.

As it is already remarked there in the comments, the equality holds when $a,b$ are coprime. Since you only ask for a reference (although the proof is easy enough), I'd recommend you the book of Herzog and Hibi, Monomial Ideals, proof of theorem 1.3.1, page 8.

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