Could someone show the reference of this rule: if $I = (ab) + J$, then $I = ((a) + J) \cap ((b) + J)$.
I have found it here.
2026-03-25 14:04:31.1774447471
Reference to this rule $I = (ab) + J$, then $I = ((a) + J) \cap ((b) + J)$
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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.